Topological Phases of Interacting Fermions in Optical Lattices with Artificial Gauge Fields

Transcription

1 Topoogica Phases of Interacting Fermions in Optica Lattices with Artificia Gauge Fieds Michae Buchhod Master s Thesis in the Department of Theoretica Physics at the Johann Wofgang Goethe University of Frankfurt Juy 2012 Supervisor and Referee: Secondary Referee: Prof. Dr. W. Hofstetter Prof. Dr. L. Bartosch

2

3 Contents 0 Introduction 1 1 The Fermi-Hubbard Mode and the DMFT Approximation The Fermi Hubbard Mode Dynamica Mean-Fied Theory Rea-space Dynamica Mean-Fied Theory Monte-Caro Methods in a Nutshe Monte-Caro Method Integration Statistica Data Anaysis for Direct Samping Procedures Markov-Chain Monte-Caro Markov-Chains The Metropois agorithm Statistica Data Anaysis for Markov-Chain Samping Procedures Diagrammatic Monte-Caro Methods From Cassica to Quantum Monte-Caro Genera Formaism of Diagrammatic Monte-Caro Methods The Sign Probem in Monte-Caro Processes The Anderson Impurity Mode and Continuous-Time Monte-Caro Methods The Anderson Impurity Mode Exact Diagonaization Continuous-Time Auxiiary Fied Quantum Monte-Caro Agorithm Configurations, Weights and Measurement Factors The Samping Procedure Fast Matrix Manipuations Benchmarking and Performance Anaysis Continuous-Time Hybridization Expansion Quantum Monte-Caro Agorithm Configurations, Weights and Observabes The Samping Procedure

4 iv Contents 4 Extensions and Improvements to the CT-QMC Methods CT-AUX on a L-site Custer Spin-Mixing Formuation of CT-AUX Superfuid Formuation of CT-AUX Spin-Mixing Formuation of CT-HYB Improvements for CT-HYB Observabes in the Legendre Poynomias Basis Direct Sef-Energy Measurement for the CT-HYB Agorithm Utracod Atoms in Optica Lattices Optica Lattice Potentias Interactions and Feshbach Resonances Hubbard Parameters for Optica Lattices Artificia Gauge Fieds for Neutra Atoms in Optica Lattices Raman Transitions in the Λ-system Zeeman Lattice pus Raman Beams Time-Reversa Invariant Topoogica Insuators with Cod Atoms Z-Topoogica Insuators in Optica Lattices The Quantum Ha Effect TKNN Invariant and Berry Phase for the QHE Topoogica Edge States and the Buk Boundary Correspondence The Hofstadter Mode Effects of Smooth Boundaries on Topoogica Edge Modes in Optica Lattices The Hofstadter Mode for Cod Atoms Edge States in Cyindrica Geometries Detection Methods Z 2 -Topoogica Insuators with Interacting Utracod Fermions Time-Reversa Invariant Topoogica Insuators Time-Reversa Invariant Hofstadter-Hubbard Mode with Utracod Fermions Hofstadter-Hubbard Mode Tunabe Magnetic Order Topoogica Phases in the Hofstadter-Hubbard Mode Herbut s Argument Concusion 123 Acknowedgements 125 Bibiography 127

5 0. Introduction During the ast century since its mathematicay correct formuation by Heisenberg, Dirac, Born, Schrödinger, von Neumann and others in the ate 1920 s, quantum mechanics has branched out into amost every aspect of 20th century physics and into many other discipines, such as quantum chemistry, quantum eectronics, quantum optics, and quantum information science. However, despite the great success of quantum mechanics in expaining the effects of nature in a mathematica rigorous way, there are sti many open issues remaining that have not yet been we understood theoreticay. These issues are commony intractabe because the physica effects can not be described by a simpified theory such as mean-fied or perturbation theory or the size of the system becomes too arge for the Hamitonian to be treated exacty by cassica computers. A semina paper by Richard Feynman [47], argued that a cassica computer wi aways experience an exponentia sowdown when it is appied to a quantum mechanica probem and proposed, as an aternative to circumvent this imitation, the concept of a universa quantum simuator. Nowadays, experimentaists have aready deveoped a poor man s version of Feynman s universa quantum simuator, which can be referred to as purpose-based quantum simuators [129]. The basic idea of this approach is to engineer the Hamitonian of the quantum system of interest in a highy controabe environment and to obtain a the desired quantities by simpy measuring its properties. A prominent exampe for a reaization of a quantum simuator is that of utracod atoms, which constitute remarkaby fexibe paygrounds for soid state and quantum many-body physics with an impressive degree of contro combined with high-fideity measurements [15, 30, 112, 26, 48]. Among the fascinating experimenta achievements of cod gases, is the reaization of an atomic Bose-Einstein condensate (BEC) [36, 5], the observation of vortices and superfuidity in a rotating fermionic gas [202] and the observation Anderson ocaization in a disordered BEC [12, 153]. Adding optica attices to utracod atom simuations, experimentaists have been abe to reaize interacting attice modes for both fermionic and bosonic atoms with a high tunabiity of the microscopic parameters [15, 60, 14], such as the interpartice interaction strength and the hopping ampitudes of the attice. With these experiments, many effects known ony from soid state physics have been expored in the context of optica attices, such as the fermionic and bosonic Mott-insuator transition [60, 98, 159] or fermionic non-equiibrium transport [73, 160]. On the other hand, many casses of systems, we known from condensed matter physics, have not yet been addressed within cod-atom experiments. This is mainy because there exist certain difficuties with their reaization, which the experimentaists have to overcome, such as very ow entropies required of the emergence of magnetic order or the absence of charge for neutra atoms, which makes it very difficut to simuate orbita magnetism. Nowadays, one direction of optica attice experiments is to impement artificia gauge fieds for neutra atoms, which mimic both the presence of an externa magnetic fied for charged partices, i.e. orbita magnetism, represented by Abeian gauge fieds, and intrinsic effects such as Rashba- or Dressehaus-type spin-orbit couping, represented by non-abeian gauge fieds. Whie there are many interesting physica regimes to address in systems with artificia gauge fieds, one major goa of

6 2 0. Introduction cod-atom experiments is to reaize topoogica phases of matter, such as quantum Ha or quantum spin Ha insuators. The genera interest in topoogica insuators began in 1980, when an unexpected quantization of the Ha conductance for a sampe in a strong magnetic fied at very ow temperatures was discovered by von Kitzing et a. [104], today known as the quantum Ha effect. The quantum Ha effect coud be expained theoreticay and concisey by Laughin ony one year ater [115] and in 1982 Haperin [77] demonstrated the existence of the famous edge states, which are gapess states at the edge of a quantum Ha sampe, energeticay connecting different buk bands, and responsibe for the quantized Ha conductance. In the same year, an infuentia paper by Thouess et a. was pubished [182], reating the quantization of the Ha conductance to the topoogy of the infinite system and therefore proving its extreme robustness against externa perturbations. This work was the starting point for the cassification of phases of matter by topoogica invariants. In the foowing years, there was a persistent interest in topoogica phases of matter eading, for instance, to the formuation of the buk-boundary correspondence and in 1988, to the introduction of Hadane s mode, which is a theoretica mode describing spiness eectrons on the honeycomb attice, which exhibits an intrinsic quantum Ha effect without the need for an externa magnetic fied [76]. In 2005, Kane and Mee [101] predicted the existence of a so-caed quantum spin Ha effect, by considering a time-reversa symmetric combination of two Hadane modes for spinfu fermions a mode that cosey resembes graphene with a strong intrinsic spin-orbit couping. This mode, today known as the Kane-Mee mode, does not show a quantized Ha conductance since spin-up and spin-down eectrons trave in opposite directions, such that the topoogica invariant for this mode is zero. On the other hand, the spin Ha conductivity is stricty quantized and Kane and Mee argued that there must be an additiona topoogica invariant, which characterizes this so-caed quantum spin Ha effect. This topoogica invariant was determined by Kane and Mee ater that year [100] to be a Z 2 invariant, taking ony the two vaues ν = 0, 1, in contrast to the quantum Ha effect, which is characterized by a Z invariant. Experimentay, the quantum spin Ha effect is very difficut to observe in graphene, since the spin-orbit couping is usuay very weak. Instead Bernevig et a. [9] predicted the quantum spin Ha effect woud appear in mercury teuride quantum wes as that compound has an unusuay strong spin-orbit couping. Ony one year ater, in 2007, the quantum spin Ha effect was observed experimentay by König et a. [110] in the proposed quantum wes. However, in two dimensions there has not been any other experiment performed that has observed Z 2 topoogica insuators in any materia. At this point, optica attice experiments with artificia gauge fieds provide the perfect payground for investigating topoogica insuators. Recenty there has been made much progress in impementing artificia Abeian [96, 140, 167, 170, 54, 97] and non-abeian gauge fieds [34, 125, 56, 120], such that the next group that experimentay reaizes a quantum spin Ha insuator coud come from the cod-atom community. In particuar, we woud ike to mention Ian Spieman s experiments at NIST, where experimentaists are attempting to synthesize a quantum spin Ha insuator with utracod 6 Li on a atom chip [57]. The controed observation of the extraordinary phenomenoogy of topoogica insuators woud shed new ight on quantum many-body theory. There are sti many open issues in the context of interacting topoogica insuators, which coud be carified by optica attice experiments and can up to now not be addressed by other soid state experiments because of the accessibe parameter regime for these is strongy imited. For instance there have been contradictory resuts pubished on the interacting phase diagram of graphene, where some Monte-Caro cacuations predict the presence of a spin-iquid phase for intermediate interactions [130, 197] whie others show the absence of this phase for any interaction strength [168], such that experimenta evidence is required for a definite answer of this probem. Further questions to address are the robustness of topoogicay protected edge states against inter-partice interactions and disorder [42, 21], the robustness of the Z 2 cassification in genera [89] or the transition from chira to heica edge states in the presence of spin-orbit interaction [56]. In this thesis, we theoreticay investigate geometry, disorder, detection methods and interactions in expicit reaizations of topoogica insuators in optica attices. The effect of the optica attice geometry, induced by the smooth trapping potentias inherent to cod-atom experiments is a major difference between optica attices and soid state systems. We investigate the topoogica invariants of these systems and show that they do not depend on the trapping geometry of the system. We show that sharp boundaries are not required to reaize quantum Ha or quantum spin Ha physics in optica attices and, on the

7 3 contrary, that edge states which beong to a smooth confinement exhibit additiona interesting properties, such as spatiay resoved spitting and merging of buk bands and the emergence of robust auxiiary states in buk gaps to preserve the topoogica quantum numbers. In addition, we numericay vaidate that these states are robust against disorder and anayze possibe detection methods, with a focus on Bragg spectroscopy, to demonstrate that the edge states can be detected and that Bragg spectroscopy can revea how topoogica edge states are connected to the different buk bands. Furthermore, we consider a spinfu and time-reversa invariant version of the Hofstadter probem which can be reaized in cod atom experiments. Using a combination of rea-space dynamica mean-fied theory and anaytica techniques, we discuss the effect of on-site interactions and determine the corresponding phase diagram. In particuar, we investigate the semi-meta to antiferromagnetic insuator transition and the stabiity of different topoogica insuator phases in the presence of strong interactions. We aso determine the spectra function of the interacting system which aows us to study the edge states of the strongy correated topoogica phases. This thesis is structured as foows: In the first section, we give a genera introduction to the Hubbard mode and introduce rea-space dynamica mean-fied theory as the numerica mode to address the physics of interacting attice modes with artificia gauge fieds. In the second section, Monte-Caro methods are introduced on a very genera eve with a focus on quantum Monte-Caro methods, namey diagrammatic Monte-Caro agorithms. For the rea-space dynamica mean-fied theory, an impurity sover is required, which soves a zerodimensiona interacting impurity probem, described by a non-interacting Green s function which is non-oca in time. In this thesis we focus on continuous-time quantum Monte-Caro impurity sovers, which are expained in great detai in the third chapter. In the fourth chapter, we give an overview of improvements and extension to the origina formuation of the continuous-time Monte-Caro sovers, which we impemented to treat systems incuding Abeian and non-abeian gauge fieds. The fifth chapter gives a brief overview over optica attice experiments and the experimenta impementation of artificia gauge fieds. To discuss the reaization of Z topoogica insuators, such as the quantum Ha effect, we begin the 6th chapter with a brief introduction of the quantum Ha effect, topoogica invariants and edge states. Then we discuss the reaization of topoogica phases in cod-atom experiments by impementing the optica attice version of the famous Hofstadter mode [87]. In the 7th chapter, we introduce Z 2 topoogica insuators, such as the quantum spin Ha effect and investigate the effect of interactions on the topoogica phase and the edge states as we as the interaction driven phase transition from a semi-meta to a magneticay ordered insuator. Subsequenty, we discuss how non-abeian gauge fieds infuence the magnetic order in the insuating phase. In the ast chapter, we provide some concusions and an outook of this thesis.

8 4 0. Introduction

9 1. The Fermi-Hubbard Mode and the DMFT Approximation In this chapter, we wi introduce the paradigmatic Hubbard mode, describing interacting Fermions on a attice and the computationa method to anayze its properties, which is the Dynamica Mean-Fied Theory (DMFT) as we as its rea-space extention, the rea-space Dynamica Mean-Fied Theory (RDMFT). Dynamica Mean-Fied Theory (DMFT) is a method we suited to anayze the oca correations of strongy interacting Fermions in homogenous attice systems in arge dimensions (i.e. d 3), and has been succesfuy used to describe the interaction driven Meta-insuator transition and to cacuate dynamica quantities of the Hubbard mode. Our interest ies in the anaysis of inhomogenous attice systems, where the inhomogeneity is caused by artificia gauge fieds and/or an additiona externa trapping potentia. We therefore make use of the extended rea space version of DMFT, which aso is described in the foowing sections. 1.1 The Fermi Hubbard Mode The fermionic Hubbard mode (more commony referred to as the Hubbard mode) is a theoretica mode, describing spinfu attice Fermions, typicay in the owest band approximation, that are interacting with each other ony ocay [91, 93, 92]. This mode, with its extensions to more genera attice structures or interactions [187, 18, 134], is one of the most important theoretica modes in condensed matter physics, being abe to describe the interaction driven meta-mott insuator transition at haf-fiing [136, 53], superconductivity in the case of attractive interactions [109, 33], inter- and intra-band magnetism [180, 178, 49] and many more effects, known from soid state experiments. In this section, we shorty review singe-partice physics in a periodic attice potentia, then introduce interactions between partices and, finay, express the corresponding many-partice Hamitonian in second quantization. At the end of the section, we give a brief overview of the physica properties of the Hubbard mode that are most reevant to the work in this thesis. Non-interacting partices in a periodic attice potentia V (x), of which we consider ony square or cubic attices, are described by the first-quantized Hamitonian H 0 = p2 + V (x). (1.1) 2m Due to Boch s theorem, the eigenfunctions of this Hamitonian can be cathegorized by two quantum numbers, namey the band index α, with α N, and the quasi-momentum k, with k x,y,z { π(2 L) La x,y,z },

10 6 1. The Fermi-Hubbard Mode and the DMFT Approximation where a x,y,z are the attice constants in the corresponding directions, L is the number of attice sites and = 1,..., L is an integer abe. The eigenfunctions of Hamitonian (1.1) are caed Boch functions ψ αk (x) and have the periodicity ψ αk (x + R) = e ikr ψ αk (x), (1.2) where R is a attice vector and fufi the eigenvaue equation H 0 ψ α,k (x) = ɛ α (k)ψ α,k (x). (1.3) The vaues ɛ α (k) for fixed α, as a function of k constitute the dispersion of the band α. In many cases, there exists a band gap α,β between the bands, i.e. α,β := min k,k ɛ α(k) ɛ β (k ) > 0, (1.4) for α β. In sufficienty deep attices, the gap between the first and second band is much arger than the bandwidth Γ of the first band, Γ = max ɛ 1(k) ɛ 1 (k ). (1.5) k,k If this is the case and additionay a other reevant energy scaes (the typica interaction strength and the temperature k B T ) are aso smaer than the first band gap, it is perfecty reasonabe to describe ow energy physics in a owest band approximation. The Hamitonian (1.1) is then reduced to H 0 = α,k α, k ɛ α (k) α, k k k ɛ(k) k, (1.6) where α, k are the Boch states and we suppress the index α = 1 in the ast step, considering ony the owest band in the Hamitonian. From a given set of Boch states { α, k } it is possibe, via unitary transformation, to construct a new set of basis states. We known in this context are the so-caed Wannier states, which are obtained from the Boch states by a Fourier transformation α, i = 1 e ikri α, k, (1.7) L k where R i is the attice coordinate of site i. The Wannier states are found to be a very good basis for expressing a genera Hamitonian H = H 0 + H as soon as H contains ony oca or short ranging operators, since the Wannier states combine both ocaization of the wave function x α, i = w i (x) on the attice site i and the separation into different bands α, being we separated by a band gap. Again, we focus ony on the owest band and reexpress the Hamitonian in the Wannier basis, which eads to H 0 = k ɛ(k) k k = i,j,k e ik(r i R j ) ɛ(k) L 2 i j = ij t ij i j. (1.8) The matrix eements t ij are caed hopping matrix eements or hopping parameters and describe the transfer rate from state j to state i in a attice. Generay, if one is interested in the interpay between a non-oca part H 0 and a oca (and often manypartice) part H in a Hamitonian H = H 0 +H and not in the exact band structure of reaistic materias, it is reasonabe to make a further approximation, caed the tight-binding approximation. In the tightbinding approximation, the hopping parameters are considered to be non-zero ony for nearest neighbours i, j in the attice. This becomes exact in the imit of very deep, isotropic attices (sometimes caed the atomic imit) and is a reasonabe approximation if the attice is not too shaow, in which case the owest band approximation woud no onger hod true. Within the tight-binding imit, the Hamitonian H 0 then finay reads H 0 = i,j t ij i j, (1.9) where i, j indicates that the sum runs ony over nearest neighbours i, j. Throughout this thesis, we wi aways make use of the tight-binding approximation, first because we are mainy interested in the

11 1.1. The Fermi Hubbard Mode 7 competing effects of oca interactions and non-oca dispersion and, second, because the periodic attice structures, which are experimentay created in optica attice experiments, are described perfecty by this approximation (see chapter optica attices). Next, we consider the representation of interactions in a attice system. In genera, n-partice interactions can be described by a sum of n-partice operators Ôn. Here, we ony consider two-partice interactions, as higher numbers of partices cannot be present on a singe attice site. Usuay, the interaction operators are diagona in rea-space, described by an interaction potentia U(x 1, x 2 ), between two partices, ocated at the coordinates x 1 and x 2. As most interaction ony depend on the distance x 1 x 2 of the corresponding partices, U(x 1, x 2 ) = U(x 1 x 2 ), which makes the interactions transationay invariant. In the Wannier representation, the interaction operator H is then given by H = U ijm ij m, with U ijm = dx 1 dx 2 U(x 1 x 2 )wi (x 1 )wj (x 2 )w (x 1 )w m (x 2 ), (1.10) ijm where ij is an anti-symmetric two fermion state, i.e. ij = 1/ 2( i j j i ). So far, we have not incuded the Fermion s spin in this formaism, which was not necessary since we do not consider interaction operators that change the spin. However, the spin is crucia for the Hubbard mode, since it has been formuated first for interacting eectrons in soid state systems. When adding a fermionic spin σ =, pointing either in positive or negative z-direction the interaction operator reads H = 1 U ijm iσjσ σmσ, (1.11) 2 ijmσσ where we expoited the fact that the spin is unchanged by the interactions. Many-partice physics can be convenienty expressed in the framework of second quantization, where the states in (1.9) and (1.11) are represented by fermionic creation and annihiation operators c iσ, c jσ. These operators fufi fermionic anticommutation reations {c α, c α } = c αc α + c α c α = δ α,α (1.12) and any anti-symmetrized fermionic N-partice state Ψ N is expressed as Ψ N = α 1...α N = N c α i 0. (1.13) Within the second quantization formaism, the Hamitonian H = H 0 + H, which is a combination of the attice Hamitonian H 0 and the interaction operator H, takes on the form i=1 H = t ij c iσ c jσ + 1 U ijm c iσ 2 c jσ c mσc σ. (1.14) ijσ ijmσσ The interaction matrix eements U ijm describe the energy associated with a scattering process of two partices in the quantum states, m, scattering into the states i, j. Very often, ony s-wave scattering processes, which are the owest order scattering processes, are considered, such that the matrix eements contributing to H up to eading order are those with i = j = = m. In soid state systems, this approximation is usuay justified by the arge screening of the Couomb interaction, which makes the resuting interaction short ranged. In contrast, in optica attice experiments, no screening effect is present but the sma energy of the system impies that p-wave scattering (and a higher orders) is unaffected by the interaction strength and can therefore be negected. With the reduction of the interaction to s-wave processes, the fina we known Hubbard Hamitonian is H = t ij c iσ c jσ + 1 U iiii c iσ 2 c iσ c iσ c iσ = t ij c iσ c jσ + ijσ iσσ ijσ i where n iσ = c iσ c iσ is the oca number operator for spin σ. Un i n i, (1.15)

12 8 1. The Fermi-Hubbard Mode and the DMFT Approximation To briefy iustrate the rich physics described by the Hubbard mode, we examine the haf-fied case for zero temperature in three dimensions. In genera, the Hubbard mode, as it is described by (1.15), has charge and spin degrees of freedom, which determine the possibe physica phases of the system. For weak interactions, the system is in a metaic phase, or Fermi iquid, which is we described by noninteracting fermionic quasipartices with an effective mass m (k) = ( 2 kɛ(k)). At a critica interaction, the charge degrees of freedom are frozen out and the system becomes stricty insuating, the caed a Mott insuator. The density of states at the Fermi energy vanishes and two separated Hubbard bands are formed, one of which is competey occupied, and the other competey empty. For stronger interactions, the system possesses ony spin degrees of freedom. For very strong interactions, the Hamitonian (1.15) can be expanded in orders of t/u, which resuts in the antiferromagnetic Heisenberg mode H AF = 4t2 U i,j ( ) t 3 S i S j + O U 2, (1.16) favoring antiferromagnetic ordering. In the imit of infinite interactions U/t the ground state of the Hubbard mode is the Née state, which is the ground state of the AF-Heisenberg mode. The excitation spectrum of the Hubbard mode can be divided into charge and spin excitations. For a regimes of t/u the spin excitations are gapess and the ow energy excitations are described in a spin-wave theory framework. In contrast, the charge excitations have a competey different structure on both sides of the critica Mott point. In the metaic phase, the charge excitations are gapess and the ow energy physics is we described by non-interacting partices, whereas in the Mott insuating phase, the excitations are gapped and the excitationa structure consists of partice- hoe excitations. With this short overview, we finish this section and come to the derivation of the Dynamica Mean-Fied Theory, the computationa method to investigate the properties of interacting attice modes, such as the Hubbard mode. The Hubbard mode itsef wi be investigated further and in more detai in the foowing chapters of this thesis in the context of topoogica phases, caused by the addition of gauge fieds to the singe-partice Hamitonian (1.9), see chapter Dynamica Mean-Fied Theory In this section, we describe Dynamica Mean-Fied Theory (DMFT) and present a derivation of the DMFT sef-consistency equations based on the cavity method. DMFT is a powerfu method, previousy used extensivey to anayze the Hubbard mode (i.e. to determine phase diagrams or to cacuate dynamica quantities) in arge dimensions d 3. The main idea of the DMFT approximation can be formuated within very few words, athough the theoretica justification of this approximation and the derivation of the sef-consistency equations is very non-trivia. Within DMFT, the sef-energy of the attice is considered to be oca, in other words the quasi-momentum dependence of the sef-energy is negected competey, Σ(k, ω) Σ(ω). (1.17) As usua in Mean-Fied theories, this projects the physica quantities (in our case the singe-partice Green s functions) onto a subspace of smaer compexity and henceforth makes these easier to determine. The chaenge of this approximation is now to find the best sef-energy of the form (1.17), which is done by appying the sef-consistency equations ater derived in this section. It was first proven by Metzner and Vohardt [133], that the sef-energy becomes oca in the imit of infinite dimensions, i.e. that the DMFT approximation is exact in infinite dimensions. Subsequenty, much progress was achieved by estabishing the DMFT approximation and justifying this approximation in finite, but arge, dimensions. One of the first achievements of the DMFT approach was then the anaysis of the interaction driven Meta-Mott insuator transition in the Hubbard mode and the determination of the exact phase diagram. To iustrate the steps towards the derivation of the DMFT sef-consistency condition, we start with writing down the action of the Hubbard mode. For theoretica cacuations, we choose to use a functiona

13 1.2. Dynamica Mean-Fied Theory 9 integra formuation in terms of Grassmann variabes as it is taught in most theoretica ectures on manybody physics. Here, we denote Grassmann variabes by c α and c α, which is the same notation as we use for fermionic creation and annihiation operators. However, in contrast to operators, the Grassmann variabes c α and c α are different and competey independent variabes. Using the same symbos for both Grassmann variabes and operators has been we estabished in the scientific iterature and it shoud aways be cear from context, which picture we are working in (i.e. operators or Grassmann fieds). In the functiona integra representation, the partition function is given by the expression Z = D[c iσ, c iσ ] e S, (1.18) where the grand-canonica action of the Hubbard mode takes on the form i,σ S = β 0 dτdτ i,j,σ c iσ (τ)(g0 ) 1 ij,σ (τ τ )c jσ (τ ) + U β 0 dτ i n i (τ)n i (τ). (1.19) Here, G 0 ij,σ (τ) is the non-interacting Green s function, as defined in the iterature, the attice sites are abeed with i, j, respectivey, and σ is the spin, σ =,, which is good quantum number for the mode considered here. Due to the oca sef-energy, in the foowing derivation, we wi see that it is sufficient to determine the oca interacting Green s functions G ij,σ (τ) to uniquey identify the matrix eements of the sef-energy. These Green s functions are determined by the the expression G iiσ (τ) = 1 D[c jσ Z, c jσ ] c iσ (τ)c iσ e S. (1.20) j,σ To simpify this expression, we reaize, that the integration over a degrees of freedom, except that of (i, σ) can be done before inserting the creation and annihiation operators in this expression. We define the effective action for the determination of the oca Green s functions by 1 e S eff,i = 1 D[c jσ Z eff,i Z, c jσ ]e S, (1.21) j i,σ such that the Green s function integra reads G iiσ (τ) = 1 D[c iσ Z, c iσ ] c iσ (τ)c iσ. (1.22) e Seff,i eff,i σ Note that the effective action S eff,iσ ony contains Grassmann variabes c iσ and c iσ and therefore, if the effective action is known, the determination of the oca Green s functions reduces to a zero-dimensiona probem (as it is for instance given by the Anderson impurity mode). Due to this, we want to find an approximate expression (again exact in infinite dimensions) for the effective action. We decompose the tota system into the oca part, i.e. the attice site i (setting i = 0 without oss of generaity) and the cavity system, which is the whoe attice but without site i and perform the same decomposition to the action of the system (1.19). This spits the action into a oca action S 0, the cavity action S (0) and the couping of the oca part to the cavity S, where we have defined S 0 = β 0 dτ S 0 (τ) = β 0 dτ σ c 0σ (τ)( τ µ)c 0σ (τ) + Un 0 (τ)n 0 (τ) (1.23) and S = β 0 β dτ S(τ) = dτ 0 jσ t j0 c jσ (τ)c 0σ (τ) + t 0jc 0σ (τ)c jσ (τ). (1.24) With this separation of terms, we can rewrite expression (1.21) such that e S eff = Z eff Z e S0 D[c iσ, c iσ ]e S e S(0) = Z eff Z (0) e S0 e S (0), (1.25) Z σ

14 10 1. The Fermi-Hubbard Mode and the DMFT Approximation where we have suppressed the index i, and introduced the partition function of the cavity Z (0) and the expectation vaue O (0) of an operator O with respect to the cavity. Reaizing that Z eff Z (0) Z is simpy a normaization factor, which contains no operators, we have found an exact expression for the oca effective action by the combination of (1.21) and (1.25), which reads S eff = S 0 n e S (0). (1.26) The first part of the effective action (the oca action S 0 ) contains terms that are of quadratic and quartic order in the Grassmann variabes, whie the second part (the averaged exponentia) contains Grassmann variabes to arbitrary arge order, which is obvious when we expand this expression into a power series. The ogarithm in (1.26) is very hepfu to bring the second part of the equation into a more convenient form using the inked custer theorem. The inked custer theorem states the foowing, supposed we have an action S and a perturbation S pert with Z = D[c iσ, c iσ ] e S Spert, Z 0 = D[c iσ, c iσ ] e S i,σ = n Z = ( ) a connected diagrams of a Z 0 perturbation expansion in S pert i,σ (1.27) Now, we can appy this theorem to the expectation vaue in Eq. (1.26) to finay obtain an expicit expression for the the effective action S eff. To understand the resut, we perform an expansion of e S = n=0 n =1 dτ S(τ ) (1.28) and take into account, that in the expectation vaue of Eq. (1.26) ony even terms of this expansion can survive, since S is inear in cavity operators c iσ, c iσ 1. Because we do not consider finite pairing fieds in this derivation, the number of Grassmann variabes c 0σ must aso be equa to the number of variabes c 0σ and we ony have to consider the respective terms in the expansion (1.28) 2. After these considerations, we may take a ook at the first non-vanishing and non-trivia summand in (1.28), which is the non-vanishing part of the second order contribution (the zeroth order is triviay the unity operator and the expectation vaue of the first order term vanishes competey), given by S 2 = 1 β dτ 1 dτ 2 c 0σ 2 (τ 1) t i0 t 0j c iσ (τ 1 )c jσ (τ 2) c 0σ (τ 2 ). (1.29) 0 σ ij Looking at Eq. (1.29), we immediatey reaize, that the missing part of the effective action consists of a sum of connected Green s functions G (0) C with respect to the cavity action. We can now exacty rewrite the effective action, which takes the form S eff = S 0 + dτ 1...dτ n dτ 1...dτ nc 0σ (τ 1)...c 0σ (τ n)m (n) C (τ 1...τ n)c 0σ (τ n)...c 0σ (τ 1), (1.30) n=1 where M (n) C (...) is the sum over a connected Green s functions G(0) C,i 1,...i n,j 1,j n (...) of order n, weighted with the corresponding hopping parameters t i1,0,...t in,0, t 0,j1,...t 0,jn, i.e. M (n) C (τ 1...τ n) = i 1,...j n G (0) C,i 1,...i n,j 1,...,j n (τ 1...τ n) n t i,0t 0,j. (1.31) Here, the number of different non-zero hopping ampitudes t i,0, t 0,j is proportiona to the dimension d and one shoud note, that the connected Green s functions from the previous expression can be reated 1 Note that for fermionic systems (even with superfuid pairing) expectation vaues of an odd number of creation and annihiation operators are aways zero. 2 This derivation can be straightforwardy extended to incude superfuid pairing of fermions but we do not show this here to avoid confusion. =1

15 1.2. Dynamica Mean-Fied Theory 11 to the fu cavity Green s functions by a hierarchy of equations, e.g. the two point connected Green s functions are identica to the fu Green s functions and for the four point Green s function, one may use the reation G C,i1,i 2,j 1,j 2 (τ 1, τ 2, τ 1τ 2) = G i1,i 2,j 1,j 2 (τ 1, τ 2, τ 1τ 2) (1.32) G i1,j 1 (τ 1, τ 1)G i2,j 2 (τ 2, τ 2) G i1,i 2 (τ 1, τ 2 )G j1,j 2 (τ 1, τ 2) G i1,j 2 (τ 1, τ 2)G i2,j 1 (τ 2, τ 1). So far, we have derived an exact expression for the effective action of an arbitrary attice site, here denoted as site 0. The effective action in Eq. (1.30) contains ony oca operators, they must be considered to infinite order and are additionay couped to very non-trivia matrix eements (those of the matrix M). From this expression we aim now to obtain a usefu approximation for the effective action by making use of the arge dimension of the system. This expression sha again be exact in the imit of infinite dimensions. From now on, we stricty foow the authors of [53] and expain their steps for the approximation. If we want to describe non-trivia physics in infinite dimensions, it is necessary to keep the average kinetic energy per partice finite (to be more precise, on the order of the interaction U, otherwise interactions can be treated as an arbitrary sma perturbation, resuting in a perfect Fermi iquid). This is achieved by rescaing the hopping parameters t i,j with the dimension. We wi now derive the proper scaing scheme to obtain a non-trivia physics in the imit of infinite dimensions as it was first expained by Metzner and Vohardt in 1989 [133]. The kinetic energy per partice ɛ kin of a d-dimensiona fermionic many-partice system at zero-temperature can be expressed as ɛ kin = dɛρ d (ɛ)f ɛf (ɛ)ɛ, (1.33) where we have introduced the density of states of a d-dimensiona system ρ d (ɛ) = α δ(ɛ ɛ α ), (1.34) with α, ɛ α abeing the eigenstates, the eigenenergies of the system, respectivey, and the zero-temperature Fermi-distribution function f ɛf (ɛ) = Θ(ɛ F ɛ), (1.35) such that dɛρ d (ɛ)f ɛf (ɛ) = n (1.36) is the partice number per attice site. When the eigenstates of the system are separabe with respect to each dimension, one can express kinetic energy per partices ɛ, which is distributed according to ρ d (ɛ), as the sum of independent variabes d ɛ = ɛ, =1 where every singe variabe ɛ is distributed independenty according to the one-dimensiona density of states ρ 1 (ɛ). Using the centra imit theorem, for the case of the dimension approaching infinity (d ), this eads to a distribution function of ρ d (ɛ) d = 1 e (ɛ ɛ) 2 dσ, (1.37) πdσ where ɛ = 0 is the average and σ = 2t is the variance of the one-dimensiona density of states. Equation (1.37) shows that the ony scaing that preserves the energy from being either infinite or zero in the imit d is to scae the hopping t according to t = Ct/ d, with a dimension independent constant C, which we choose C = 1. For any other choice of the scaing, the density of states woud describe a mode with an infinite kinetic energy, i.e. a purey metaic system, or with zero kinetic energy, i.e. a system in the atomic imit. The dimension dependent scaing of the hopping parameters introduced above, i.e. t t/ d, wi aso

16 12 1. The Fermi-Hubbard Mode and the DMFT Approximation ead to a specific scaing behavior of the Green s functions G ijσ (ω) with the dimension d. The Green s functions are defined as the matrix eements of the resovent operator 1 G ijσ (ω) = iσ R ω jσ = iσ ω H jσ = 1 ω iσ ( ) n H jσ, (1.38) ω whereas we used the definition of the Neumann series in the ast equaity 3. For off-diagona matrix eements i j = 0 of the Green s functions, the owest order non-zero matrix eements in equation (1.38) are those with n =, which are at east proportiona to t and therefore scae as d /2. This means, in the imit of arge dimensions, the singe-partice Green s functions wi scae as n=0 G ijσ d 1 2 Ri Rj. (1.39) This resut is quite important, not ony because it wi be used further on to anayzed the connected Green s functions in (1.31) but aso because it eads to the fact that in infinite dimensions, the sef-energy becomes a purey oca quantity, which is the essence of the DMFT approximation. Within diagrammatic perturbation theory in the oca interaction U, the sef-energy is defined as the sum over a connected, one-partice irreducibe diagrams. With the scaing in Eq. (1.39), a contributions to the sef-energy that contain non-oca Green s functions wi vanish in infinite dimensions and the sef-energy becomes a purey oca quantity Σ ijσ (ω) = δ ij Σ iiσ (ω), (1.40) as we stated in the beginning of this chapter 4. Before we proceed in rescaing the effective action of Eq. (1.30), we have to consider the scaing of the connected Green s functions G (0) C,i 1,...j n. As pointed out in [53], these scae as G (0) C,i 1,...j n n i 1 i i 1 j d d. (1.41) =1 With this, the sum in (1.30) can be further anayzed. For a given order n, in this sum there wi aways contribute d m summands with m distinct indices, 1 m 2n. If two indices of (1.41) are distinct, their distance is at east two attice sites, which eads to a scaing for the Green s functions of G (0) C,i 1,...j n d m 1, if m indices are distinct. The tota scaing of the nth order contribution is given by d n from the 2n rescaed hopping ampitudes, mutipied with d m d 1 m = d coming from the Green s functions and therefore in tota amounts to d 1 n. In the imit of infinite dimensions d, the ony non-vanishing term in the sum of Eq. (1.30) is the first order term. With this approximation, the effective action finay reads S eff = S 0 + dτdτ c 0σ (τ)t i0t j0 G (0) ijσ (τ τ )c 0σ (τ ) (1.42) ijσ and by defining the hybridization function Γ σ as Γ σ (τ) = ij t 0i t j0 G (0) ijσ (τ), (1.43) the effective action is given by S eff = dτdτ c 0σ (τ) (δ(τ τ )( τ + µ) Γ σ (τ τ )) c 0σ (τ ) + U σ dτ n (τ)n (τ). (1.44) In this effective action, the hybridization function Γ σ describes the first order couping term of a singe attice site to an interacting attice probem, where the respective attice site has been removed, of an 3 The states iσ denote Wannier states at attice site i for spin σ and we omitted the convergence factor +i0 + in the resovent operator. 4 At this point, one shoud be aware that this is ony true for oca interactions or iterations that are of density density type but not for non-oca exchange interactions

17 1.2. Dynamica Mean-Fied Theory 13 expansion in the dimension d of the system, whereas a higher order terms scae at east as O(d 1 ). We now introduce one further quantity, the so-caed Weiss function G 0, which compactifies the notation of the effective action. We define G 1 0σ (τ) = δ(τ)( τ + µ) Γ σ (τ), (1.45) and insert this into the effective action S eff = dτdτ c 0σ (τ)g 1 0σ (τ τ )c 0σ (τ ) + U σ dτ n (τ)n (τ). (1.46) With this ast equation, the effective action derived in this section describes a zero dimensiona spinfu and interacting system, where the Weiss function has taken over the part of the non-interacting Green s function of this system and the interaction is described by the Hubbard U. From now on, we wi ca this zero dimensiona system the impurity system or simpy the impurity. Importanty, one shoud not be confused by the Weiss function, since it is the non-interacting Green s function of the impurity system, but it is not the oca projection of the non-interacting attice Green s function. As mentioned before, the hybridization function describes the couping of the impurity system to an interacting attice probem and therefore this information is aso contained in the Weiss function. Within the derivation of the effective impurity mode, we can identify two important quantities, that coincide for both the attice and the impurity probem. As we mentioned at the beginning, the reason for deriving an effective action of the above form (1.46) is that we wanted to find a very compact expression for the oca interacting Green s functions of the attice probem as given by (1.20) and (1.22). Looking at these two equations, we can immediatey see, that within the DMFT approximation the interacting attice Green s function G iiσ (τ) exacty coincides with the interacting Green s function G 0σ (τ) of the impurity probem. Later, we wi demonstrate that the oca attice sef-energy is aso identica to the sef-energy of the impurity probem, which competes the DMFT sef-consistency equations. In the ast part, we have introduced the DMFT approximation, which resuted in the insight that the sef-energy of a attice probem with oca interactions (as it is the case for the Hubbard mode) becomes purey oca in infinite dimensions and that the non-oca contributions scae at east as O(d 1 ) and aso that the effective action, sufficient to describe a oca correations, takes on the form of a zero dimensiona interacting probem, caed the impurity. As we wi see in the foowing parts, this is aready sufficient to, without any further approximations, construct a fuy sef-consistent theory to approximatey describe the interacting attice probem in arge dimensions, becoming exact in the imit of infinite dimensions. The physica systems investigated in the ater parts of this thesis do not have the transationa symmetry of an infinite attice and therefore aways require a rea space anaysis of a finite system, which was done by using RDMFT, the rea space extension of DMFT. To derive the sef-consistency equations used in RDMFT is is sufficient to start with a the reations that we have derived so far and therefore the reader may directy jump to the RDMFT section now. However, for competeness, we wi aso derive the DMFT equations which have been extensivey used to investigate a homogeneous attice system in the thermodynamic imit. This wi be done in the residua part of this section but is, as mentioned before, neither necessary to understand the RDMFT equation nor the physics that are presented ater in this thesis. To derive the fu set of DMFT equations for transationa invariant systems, we precisey foow the steps iustrated in [?]. We start with switching from imaginary-time to Matsubara frequency representation, where β f(τ) = f(iω n )e iωnτ and f(iω n ) = dτf(τ)e iωτ (1.47) n= is the transformation between these two representations for an arbitrary function f and ω n = (2n + 1)π/β are the fermionic Matsubara frequencies. In this representation the hybridization function is given by Γ σ (iω n ) = ij t 0i t j0 G (0) ijσ (iω n) = ij 0 ( t 0i t j0 G ijσ (iω n ) G ) i0σ(iω n )G 0jσ (iω n ), (1.48) G 00σ (iω n )

18 14 1. The Fermi-Hubbard Mode and the DMFT Approximation where the ast equaity can be proven for any attice, with one attice site removed, and can for instance be derived by performing an expansion of the Green s functions in the hopping parameters t, as mentioned in [53]. To evauate the sum in Eq. (1.48) we express the Green s functions in quasi-momentum representation G ijσ (iω n ) = 1 e ik(r i R j ) G kσ (iω n ) = 1 e ik(r i R j ) 1, (1.49) L L iω n + µ Σ σ (iω n ) ɛ k k where we aready used the k independence of the sef-energy Σ kσ (iω n ) = Σ σ (iω n ). The dispersion ɛ k is nothing ese but the Fourier transformation of the hopping parameters k ɛ k = i t 0i e ikri, (1.50) which eads, when combined with (1.48), to Γ σ (iω n ) = I 2 I2 1 I 0, (1.51) where we introduced the different sums for convenience I 0 = 1 1, (1.52) L ξ ɛ k I 1 = 1 L I 2 = 1 L k k ɛ k ξ ɛ k = 1 L ɛ 2 k ξ ɛ k = 1 L k k k ( ɛk ξ + ξ ) = 1 + ξi 0, ξ ɛ k ξ ɛ k (1.53) ( ɛk (ɛ k ξ) + ξɛ ) k = 0 + ξi 1. ξ ɛ k ξ ɛ k (1.54) In these equations, we have used the shortened notation ξ iω n + µ Σ(iω n ) and made use of the trivia identities 1 = 1 and ɛ k = 0. (1.55) L k k Now, with a itte bit of agebra, we can rewrite the hybridization function so that it is ony in terms of I 0, which is simpy the interacting impurity Green s function G 0σ (iω). We obtain Γ σ (iω n ) = iω n + µ Σ σ (iω n ) G 1 0σ (iω n), (1.56) which, after identifying the Weiss function in Matsubara representation as G0σ 1 (iω n) = iω n +µ Γ σ (iω n ) finay reads G 1 0σ (iω n) = G0σ 1 (iω n) Σ σ (iω n ). (1.57) The ast equation is nothing but the Dyson equation for the impurity probem, which means that not ony the interacting oca attice Green s function is identica to the impurity Green s function but aso the oca attice sef-energy is identica to the impurity sef-energy. This is a remarkabe resut, especiay as no further approximations were necessary in the derivation of these identities. This ast identity (1.57) competes the set of four DMFT equations required for the transationa invariant attice. To concude this section, we shorty review these four equations and expain the DMFT sef-consistency procedure. We started by deriving a oca effective action, which becomes exact in infinite dimensions and reads S eff = dτdτ c 0σ (τ)g 1 0σ (τ τ )c 0σ (τ ) + U dτ n (τ)n (τ) (1.58) σ and can be used to determine the oca correation functions such as the oca interacting Green s function G iiσ (iω n ). During this derivation, we argued that the sef-energy becomes purey oca in infinite dimensions, which was rigorousy proven by Vohardt and Metzner [133], i.e. Σ kσ (iω n ) = Σ σ (iω n ), (1.59)

19 1.3. Rea-space Dynamica Mean-Fied Theory 15 which eads to the simpified attice Dyson equation G 00σ = 1 L k 1 iω n + µ Σ σ (iω n ) ɛ k. (1.60) The ast important equation is the identification of the impurity sef-energy with the oca attice sefenergy, which is best expressed through the impurity Dyson equation G 1 0σ (iω n) = G 1 00σ + Σ σ(iω n ). (1.61) If the equations (1.60) and (1.61) are fufied simutaneousy by a Green s function G and sef-energy Σ, which resut from the impurity action (1.58), a sef-consistent soution of the DFMT equations is found and G, Σ are approximate soutions for the interacting attice probem, which become particuar good in arge dimensions, i.e. d 3. A typica DMFT cacuation wi start with a guess of the hybridization functions Γ σ (iω n ) and impement a sover for the non-trivia probem of cacuating the impurity Green s function from (1.58). Then iterating the four DMFT equations (1.58), (1.59), (1.60), (1.61) wi aow convergence to the sef-consistent soution. 1.3 Rea-space Dynamica Mean-Fied Theory In the previous section, we introduced the DMFT approximation for a certain cass of interacting attice systems and, in the second part of the section competed the set of required DMFT equations for a homogenous attice (1.61) by identifying the oca attice sef-energy with the impurity sef-energy. In this section, we wi introduce the rea-space extension of DMFT, caed rea-space dynamica mean-fied theory (RDMFT), which can describe inhomogenous attice systems of finite size, caused for instance by a trapping potentia, and wi aso ater be used to anayze systems of infinite size with a arger unit ce than a singe attice site. Recenty, RDMFT has been succesfuy appied to both, disordered systems, anayzing the competing phases of meta, Mott-insuator and Anderson-insuators [163, 164, 40, 135], as we as finite systems, which were subjected to a confining potentia in rea-space [147, 166, 59, 84]. The systems that we investigate are described by the Fermi Hubbard mode with an inhomogenous distribution of on-site energies ɛ i and non-uniform hopping parameters t ij, i. e. the Hamitonian H = ) (t ij c jσ c iσ + h.c. (µ ɛ i ) n iσ + U n i n i. (1.62) ijσ iσ i This Hamitonian was previousy used to describe disordered systems [39, 179, 1, 163], where the {ɛ i } were distributed randomy, or trapped, finite systems as they appear naturay in cod atom experiments. In the foowing chapters, we wi investigate this Hamitonian with {ɛ i } representing a trapping potentia in the shape of a hard-wa confinement. This wi be done to resove the so-caed edge states of the system, which are not present in unconfined systems, described by the Hubbard mode for the homogenous case. One can aso justify the use of RDMFT instead of DMFT from a physica point of view. DMFT was formuated for a homogenous system, therefore whenever thermodynamic phases appear that break the transationa symmetry of the attice, the formuation of DMFT in the previous section wi not be abe to resove those, simpy because it is restricted to homogenous systems. In the cases where transationa symmetry is not competey broken but instead the unit ce of the system is just enarged to two attice sites (as for instance for an antiferromagneticay ordered phase), it is possibe to simpy use the origina DMFT equations with a modified sef-consistency condition. However, if the system is in a attice symmetry broken phase with hugey enarged unit ce or even without any transationa symmetry remaining (as for instance in an Anderson insuating phase), it is necessary to find a theoretica approach that is abe to resove this effects in rea-space, which is the case for the RDMFT approximation. To derive the RDMFT equations, we start with introducing the Dyson equation for a attice system in rea-space. The non-interacting Green s function in a rea-space formuation is determined by the equation (Ĝ0 σ) 1 (iω n ) = (µ + iω n ) ˆ1 ˆT ˆV, (1.63)

20 16 1. The Fermi-Hubbard Mode and the DMFT Approximation where ˆ1 is the unity operator, ˆT is the hopping operator, i.e. i ˆT j = t ij 5, and ˆV is the operator with the on-site energies of the corresponding attice sites, i ˆV j = δ ij ɛ i. With this equation, we directy come to the Dyson equation in rea-space, which reads Ĝ 1 σ (iω n ) = (Ĝ0 σ) 1 (iω n ) ˆΣ σ (iω n ) = (µ + iω n ) ˆ1 ˆT ˆV ˆΣ σ (iω n ), (1.64) with the sef-energy operator i ˆΣ σ (iω n ) j = Σ ijσ (iω n ). In this equation, we have made use of the fact that, up to now, spin is a good quantum number, since the Hamitonian (1.62) contains no terms that change the spin of a partice. When we ater anayze systems where a singe partice s spin is not onger conserved, it is straight forward to extend (1.64) to incude these terms and simiary for the derivation of the RDMFT equations. Now, as in the DMFT section, we want to derive an effective action S (i) eff, which is abe to determine the oca Green s function on attice site i, via G iiσ (τ) = 1 Z (i) eff σ D[c iσ, c iσ ]c (i) iσ (τ)c iσ e S eff. (1.65) Fortunatey, the derivation of the effective action and aso of the equations (1.48) and (1.40) in the DMFT section have reied ony on the fact that the Hamitonian was formuated on a attice but not on the transationa symmetry of the attice Hamitonian itsef (this fact just was expoited in the derivation of the ast DMFT equation (1.57)). Therefore, we can aready define the effective action S (i) eff as S (i) eff = dτdτ σ where the Weiss function G 1 iσ is defined as c iσ (τ)g 1 iσ (τ τ )c iσ (τ ) + U dτ n i (τ)n i (τ), (1.66) G 1 iσ (τ) = δ(τ)( τ + µ ɛ i ) Γ iσ (τ). (1.67) The hybridization function Γ iσ (iω n ) in Matsubara representation is determined, as in the previous section, by Γ iσ (iω n ) = t 0 t m0 G (0) mσ (iω n) = ( t 0 t m0 G mσ (iω n ) G ) iσ(iω n )G imσ (iω n ). (1.68) G iiσ (iω n ) m m The sef-energy Σ ijσ (iω n ) is again purey oca Σ ijσ (iω n ) = δ ij Σ iiσ (iω n ) but, in contrast to the previous section, may vary from attice site to attice site and, due to this, possess an additiona dependence on i. The missing step to a cosed sef-consistency oop, as in the homogenous case, is to reate the impurity sef-energy Σ (imp) σ (iω n ) = G 1 iσ (iω n) G 1 iiσ (iω) to the attice sef-energy Σ iσ(iω n ). Now, we define as the diagona (in rea-space) part of Eq. (1.64), such that ˆξ σ (iω n ) = (iω n + µ) ˆ1 ˆV Σ σ (iω ˆ n ) (1.69) Ĝ 1 σ (iω n ) = ˆξ σ (iω n ) ˆT. (1.70) For the next few ines, we drop the spin index and Matsubara frequency argument since a operators are diagona with respect to these and this simpifies the cacuation. Rewriting (1.68) then eads to Γ i G ii = m t 0 t m0 (G m G ii G i G im ), (1.71) which we express in matrix notation, expoiting the fact that ˆT = ˆξ Ĝ 1, Γ i G ii = G ii i ˆT Ĝ ˆT i i ˆT Ĝ i i Ĝ ˆT i = G ii i (ˆξ Ĝ 1 )Ĝ ˆT i ( i ˆξĜ i i (ˆξ Ĝ 1 )Ĝ i i Ĝ(ˆξ Ĝ 1 ) i ) ( ) = G ii i ˆξĜ(ˆξ Ĝ 1 ) i 1 i Ĝˆξ i 1 = G 2 iiξ 2 i G ii ξ i (G 2 iiξ 2 i 2G ii ξ i + 1) = G ii ξ i 1. (1.72) 5 Here, to be precise, j abes a singe partice state, describing a partice in the Wannier state w j.

21 1.3. Rea-space Dynamica Mean-Fied Theory 17 This equation can be rearranged and, after reintroducing spin and Matsubara frequency, reates the hybridization function of the impurity to the attice sef-energy and Green s function. Expicity, we find G iiσ (iω n ) = 1 iω n + µ ɛ i Γ iσ (iω n ) Σ iσ (iω n ), (1.73) which reates the diagona parts of attice Green s function G iiσ and attice sef-energy Σ iiσ to the Weiss functions G iσ (iω n ) = iω n + µ ɛ i Γ iσ (iω n ) via G 1 iiσ (iω n) = G 1 iσ (iω n) Σ iσ (iω n ). (1.74) With this ast equation, we again found a set of cosed equations which describe an ihomogenous interacting attice system in the imit of arge dimensions. The short interpretation of RDMFT is as foows, the equations (1.74) and (1.66) describe a set of L independent, interacting impurities, corresponding to L attice sites, which are then reated to each other by the attice Dyson equation (1.64) and the approximation of a purey but spatia dependent oca sef-energy. At ast, we now summarize the numerica procedure to obtain a sef-consistent soution of the above equations. Given a set of parameters {µ, ɛ i, t ij, U}, one foows the iterative scheme: Start from a set of initia hybridization functions {Γ iσ (iω n )} to obtain the set of effective actions }, one for each impurity. {S (i) eff For every effective action S (i) eff, cacuate the interacting Green s function G iiσ (iω n ) by soving the impurity probem (see chapter AIM). Obtain the oca sef-energies, by appying the impurity Dyson equation (1.74) at each site and subsequenty insert these into the attice Dyson equation (1.64). Invert the attice Dyson equation to obtain new oca Green s functions G iiσ (iω n ). Finay, determine a new set of hybridization functions Γ iσ (iω n ) by appying using Eq.(1.73) and start again with the first point. This procedure then is repeated unti convergence of the hybridization functions (or sef-energies) is reached. During the iteration, the computationay demanding steps are the inversion of the attice Dyson equation, scaing with O(L 3 ), where L is the number of attice sites and to cacuate the oca Green s functions as the soutions of the impurity probem. If a good spectra resoution of the Green s functions is required, soving the impurity probem using approximation schemes (which are aready quite computationay demanding) is not sufficient and we have to use exact methods, as for instance Numerica Renormaization group or Quantum Monte-Caro methods, see chapter AIM. In the foowing chapters, we wi extensivey use RDMFT to obtain the sef-energies, as we as the spectra functions of interacting Fermions subjected to artificia gauge fieds, that turn the hopping parameters compex and spatiay dependent and investigate both the infinitey extended as we as finite, trapped systems. Originay, RDMFT was introduced to sove the disordered attice probem with interactions, as done by Dobrosavjevic et a. [40, 135], as we as by Semmer et a. [163, 164,?], to improve the understanding of Anderson ocaization with interactions compared to simpified, effective theories. More recenty, it has been succesfuy used in the context of trapped attice systems, since the actua experiments with utracod atoms in optica attices naturay bring aong a trapping potentia, strongy infuencing the sma systems at hand [147, 166, 59, 84].

22 18 1. The Fermi-Hubbard Mode and the DMFT Approximation

23 2. Monte-Caro Methods in a Nutshe Monte-Caro is a generic name for a huge cass of computationa methods, which are widey used not ony in natura sciences but aso extensivey in other fieds ike for instance economy. Because of this and, of course, due to its great success in simuating computationay difficut dynamica processes, Monte-Caro methods are the most we-known computationa methods in modern times. Athough Monte-Caro methods in genera are we-known, few are intimate with the concept and working principe behind the method. This might be caused by the fact that today s Monte-Caro methods are strongy optimized to capture the hardest probems with the east computationa effort and may not be understandabe for non-speciaists. However, as we wi see in the foowing sections, the idea of Monte- Caro methods (referred to as MC) is very simpe and the straight-forward impementation of a Quantum Monte-Caro (QMC) agorithm (without care for maximizing performance) is not too difficut. 2.1 Monte-Caro Method Integration As an introduction to MC, we start with the idea of Monte-Caro integration. Suppose one simpy wants to numericay compute the integra I n (f) = dx n f(x) (2.1) Ω of the function f : Ω R, with Ω R n being a n-dimensiona area. Using the standard Simson quadrature procedure the numerica error ( ɛ scaes ) with the number of points N, on which the function is expicity evauated, with the order of O N 4 n. In contrast, if one picks the N points {x i Ω i = 1,..., N} randomy but uniformy distributed and then evauates the sum J(f, N) = 1 N N f(x i ) = f u, (2.2) i=1 where f u indicates samping of f with respect to a uniform distribution, the error is aways ony of statistica nature and therefore ( ) J(f, N) I n (f) = O N 1 2 (2.3) is competey independent of n. This procedure is caed statistica samping of the function f and wi converge to the desired integra simpy because of the aw of arge numbers. The scaing of the statistica error is expained in detai in section 2.1.1, however for the moment take (2.3) as a given property of the

24 20 2. Monte-Caro Methods in a Nutshe method. What can we deduce from this simpe exampe? From the above equations, we see that, from a scaing point of view, performing statistica samping of f gives a much better error than Simpson quadrature of f, as soon as the dimensionaity of the probem n is arger than 8. However, concuding from this resut that statistica samping is aways better than Simpson quadrature woud be a dangerous faacy since the scaing of the error does not te anything of the error itsef. A we know is that for N sufficienty arge, the error of both methods is given by Simpson (N) = 0,Simpson N n 4 and Samping (N) = 0,Samping N 1 2, (2.4) with the two unknown prefactors 0,Simpson and 0,Samping. Athough the scaing of the statistica samping is very favorabe, one expects 0,Simpson << 0,Samping because the structure of f enters the Simpson quadrature but not the statistica samping procedure. Hence a very arge number N of points may be needed in statistica samping for reasonabe accuracy and this number might be much arger than that is needed for a sufficienty sma error within Simpson quadrature and therefore statistica samping is not a priori superior to Simpson. Athough 0,Samping is unknown, we can seek to improve it without changing the scaing of the error, by using importance samping rather than statistica samping, which takes the structure of f into account for the samping process. To understand the concept of importance samping, we iustrate it by a simpe exampe. Suppose the function f is non-zero ony on a sma part of Ω, then the statistica samping procedure wi pick many points x Ω with f(x) = 0 and therefore wi waste ots of steps by choosing zeros. The best method here woud be to ony pick those points x with f(x) 0, again randomy but with certain probabiities, which are suitabe for the function f. The idea of importance samping is then to focus the samping procedure on regions in Ω, where f shows its main features and with this procedure decrease the prefactor of the error 0,Samping. Formay, we can express this idea by rewriting the integra (2.1) into I n (f) = dx n f(x) = dx n p(x) f(x) p(x), (2.5) which is formay exact. If p(x) is a probabiity density, i.e. p(x) 0, x Ω and Ω dxn p(x) = 1, we can moduate the samping procedure by picking N points x i Ω with the probabiity p(x i )dx and then compute the sum J(f, N) = 1 N N i=1 f(x i ) p(x i ) = f, (2.6) p p where F p indicates the samping of F with respect to the probabiity density p. The scaing of the error of the samping methods wi not be infuenced by this change of the method since it resuts from the samping of independent stochastic variabes which is the case for both statistica samping and importance samping. On the other hand, one now expects a much smaer prefactor of the error than for the bare statistica samping. The chaenge of the importance samping is now to chose the best probabiity density p since it shoud both mimic the important detais of f to ower the prefactor in the error but, of course, it must aso be easy to hande in a computationa simuation. For instance one coud chose ( 1 p(x) = f(x) dx f(x)) n, (2.7) which woud mimic f(x) perfecty, but in order to find this density p, one must have aready soved the fu probem aready. We now generaize the idea of importance samping to a arger cass of integras. For this, we consider the functions w, f : Ω R, having identica support (i.e. they are non-zero on the same cosed subset Ω Ω, which we refer to in the foowing as Ω), where Ω is now an abstract set, for instance composed of continuous and aso discrete, bounded and aso unbounded variabes that we ca configurations C Ω. As an exampe, reevant to the cacuations in this thesis, this set coud be given as {(n, s, τ 1,..., τ n ) n N 0, s = ±1, τ i [0, β]}, which is a mixture of a the above mentioned cases. For this set, we can define a measure (in the strict mathematica sense) µ C, which usuay coincides with the generaized Lebesgue measure (however in some cases one has to use the Haar measure), such that integras ike dµ C w C (2.8) Ω

25 2.1. Monte-Caro Method Integration 21 are we defined and the imit theorems from Lebesgue measure theory appy 1. The expicit measure wi never be important throughout this thesis and we wi shorten the integra expression by setting dc dµ C. For the above exampe of our configuration space Ω, we can write down the integra as n β dc = dτ. (2.9) Ω n=0 s=±1 =1 Now, the integras that we are interested are of the form I(w, f) = Ω dcf C Ω Ω dcw = dcw Cg C C Ω dcw, (2.10) C where we have defined 2 g C = f C w C. The mapping w C is caed the weight function, such that the vaue of w C for a certain configuration C is the respective configuration s weight. Assuming w C 0, C, we can define the probabiity of a configuration p C through w C 0 p C = Ω dc w. (2.11) C The integra I(w, f) can then be soved by importance samping of g with respect to p, where g is a function of (f, w) and p is a function of w, as defined above, i.e. the sum J(w, f, N) = 1 N N i=1 g Ci =: f MC w (2.12) converges to the integra in the imit of N. In (2.12), we have introduced a new notation. The expression f MC w, is shorthand for the samping of the function f with respect to the weight function w, as we have introduced it in the above formaism. The generaization introduced in the atter paragraph does, on the first view, not seem to simpify the probem of integration of f and even more it is uncear if one can ca this importance samping. The weight function w at this point is arbitrary and a bad choice of w wi make the samping process even worse compared to statistica samping. However, probems of the form (2.10) are standard probems that appear in statistica physics and the weights w C are the weights of certain physica configurations C. In other words w C indicates if a certain configuration pays an important roe in the integra or not. Therefore, with the regard on foowing appications on physica probems, we ca aso ca this importance samping. Now, there exist two possibiities: If the integra dc w C is known (or can be computed exacty), then one can directy cacuate the probabiities p C and start with the importance samping of f (this wi never be the case in this thesis). The integra dc w C can not be computed exacty (sma errors may aready strongy infuence the samping of f), so one needs to use further toos from stochastics and introduce Markov-Chain Monte-Caro samping, as we show in the foowing chapter. In this section, we have introduced statistica and importance samping, which are both direct samping methods, to evauate compicated integras in arbitrary dimensiona spaces. We have argued that samping procedures have a better scaing behavior with the number of step points for arge dimensions than Simpson quadrature (and aso a other numerica integration procedures). In the next section, we wi introduce a itte bit of statistica data anaysis, to understand the error of the samping procedure, which is of statistica nature. In the next chapter, we wi then introduce Markov-Chain Monte-Caro (MCMC), which can be appied to a more genera set of probems, where the integra over the weight functions does not have to be computed. 1 Our intention here is not to fuy cover the mathematica subteties of stochastic anaysis but to introduce a mathematica exact formuation of the probem such that a rigorous mathematica treatment can be appied without any obstaces. 2 This is we-defined since both functions have the same support.

26 22 2. Monte-Caro Methods in a Nutshe Statistica Data Anaysis for Direct Samping Procedures In the previous section, we have introduce direct samping procedures as a too to numericay compute compicated integras on arbitrary dimensiona spaces. In this sma section, we wi briefy discuss the error estimation within samping procedures, which are of statistica nature and therefore not ony numericay but aso fundamentay different from errors that come from anaytica approximations. Again, we formuate the probem in a genera space, denoted by Ω with configurations C Ω. On this space, we consider a probabiity density p : Ω R +, with vaues p C and the normaization dc p C = 1. (2.13) Ω Additionay, we consider a mapping (which we wi assume is a function) f : Ω R without any further features. The mean and variance of f with respect to the probabiity density p are defined as f p = dc p C f C, (mean), (2.14) Var(f) = (f f p ) 2 = Ω Ω dc p C (f C f p ) 2 = f 2 p f 2 p, (variance). (2.15) From now on we wi skip the index p and the integration space Ω. As a direct consequence of these definitions, we can derive the fundamenta reations for mean and variance af + b = a f + b and Var(af + b) = a 2 Var(f), (2.16) for any rea parameters a, b. An important concept in the context of stochastics and Monte-Caro methods is the concept of independent random variabes. In our case, f is a random variabe, which we can shorty define as: f takes vaues ξ R with probabiity p(ξ) 3. Consider now another random variabe g which is a mapping g : Ω R, where Ω may differ from Ω. The eements C Ω are aso caed configurations and have a certain probabiity density p C. Then, g and f are caed independent random variabes if and ony if P (f = ξ, g = χ) = P (f = ξ) P (g = χ) = p(ξ) p (χ), (2.17) where P (f = ξ, g = χ) is the probabiity that f takes the vaue ξ and at the same time g takes the vaue χ. In other words, for a certain reaization of configurations, if we know the vaue of g, we have absoutey no information on the vaue of f and the other way round. For independent variabes f, g, the variance of their sum Var(f + g) is simpy the sum of their variances Var(f)+Var(g), since Var(f +g) = (f +g) 2 (f +g) 2 = (f 2 +2fg+g 2 ) f 2 2 f g g 2 = f 2 f 2 + g 2 g 2. (2.18) In the second step, we made use of the fact that the expectation vaue is inear in its components and in the third step, we identified fg with f g, which is the case for independent variabes. We now define the random variabe F N which is the sum F N = 1 N N f i (2.19) of N pairwise independent random variabes f i, but with the same probabiity density. To bring this into our picture from the previous section, there now exist N identica copies Ω i with possibe configurations C i and the f i are functions f i : Ω i R, whereas the configurations C i, C j are competey independent for i j. On the other hand, if (by chance) C i = C j for any reaization, then f i,ci = f j,cj. This directy eads to the two foowing, very important resuts i=1 F N = 1 N N N i=1 f i = f i = f i, (2.20) N i=1 3 Pease note that f has not necessariy to be injective for this definition. There wi aways exist a we-defined subspace Ω ξ Ω with f(ω ξ ) = {ξ}. Then we define p(ξ) = Ω ξ dc p C.

27 2.1. Monte-Caro Method Integration 23 Var(F N ) = Var( 1 N N i=1 f i ) = 1 N 2 N i=1 Var(f i ) = Var(f i) N. (2.21) Here, we made use of the fact that the f i are independent and identicay distributed. On the right side of the two equations it is therefore not important which of the N f i is inserted in the mean or variance. The random variabe F N has favorabe properties, its expectation vaue is independent of N and equas the expectation vaues f i, where the f i are by definition identica copies of f and therefore the expectation vaue of F N is the desired integra of f over the compete space Ω, with probabiity density p. On the other hand, the variance of F N decreases with 1/N, which make F N sharpy peaked around its expectation vaue, in contrast to f itsef, which may be of arbitrary structure. The question is now, how to reaize F N. Fortunatey, this question has aready been answered if one takes a ook at the definition of J(f, N) in the previous section (2.6). As one directy reaizes, J(f, N) is identica to F N and a given reaization of J(f, N) is therefore nothing ese but a given reaization of F N. In the ast paragraph, we have derived that the variance of J(f, N), as we were using it in the previous chapter, is simpy given by Var(J(f, N)) = Var(f) N. (2.22) Now, we have to understand how this is reated to the error of J(f, N) at the end of a numerica simuation and how this error has to be understood. For this, we take a ook at Chebyshev s inequaity, for an arbitrary random variabe x with probabiity density p Var(x) = (x x ) 2 p(x) dx (x x ) 2 p(x) dx ɛ 2 p(x) dx. (2.23) Ω x x ɛ x x ɛ In this equation, the ast integra is nothing but the probabiity p( x x ɛ). Since this resut is genera, we can do the same for J(f, N) to find { } probabiity that > 1 Var(f) J(f, N) f > ɛ N. (2.24) In contrast to non-statistica methods, where the error resuts from approximations and can be made in principe arbitrary sma, the error here is of probabiistic nature and can, by coincidence, be arbitrariy arge for a particuar reaization of random variabes, this is just very unikey. The nice resut of (2.24) is that one is even abe to tune the probabiities by hand, just by changing the number of steps N in the samping procedure. Nevertheess, it has to be pointed out that the error that we are taking about is a probabiistic one and therefore aways must be handed with care. For instance, reying resuts on a singe simuation, athough deviations from a sma region around the expectation vaue may be very unikey, is absoutey dangerous and gives absoutey no contro of the error or the resut. One may ask now, so what is the error of direct samping procedure? As mentioned above, since we are deaing with probabiities there is no direct answer to this question. However, it is estabished that one can take the standard deviation of the samping procedure σ(j(f, N)) which is the square root of the variance σ(j(f, N)) = Var(J(f, N)) = σ(f) N (2.25) as a measure for the error in terms of probabiities. With a probabiity of around 68% the resut of a numerica samping wi be in the interva [ f σ, f + σ] and with a probabiity of around 95% the resut wi be in the interva [ f 2σ, f + 2σ], which is a resut from the centra imit theorem, that we wi not derive in this thesis but is expained in many books on stochastics and is aso known from the basic experimenta courses on data anaysis. Therefore the error (or what we convenienty ca the error) is proportiona to σ(f) N, which shows us the scaing with N, as we have used it in the previous section and aso expains the advantage of importance samping. The factor σ(f) strongy depends on the probabiity distribution with which the samping is performed and changing this distribution may change the standard deviation σ by many orders of magnitude 4. 4 An important factor at this point is the size and geometry of Ω. For instance, if Ω is very arge, i.e. Ω dc >> 1 the change of σ wi be significant.

28 24 2. Monte-Caro Methods in a Nutshe In this section, we have derived an expression for the statistica error in a direct samping process, as previousy introduced, which can ony be expressed in terms of probabiities. The most convenient and estabished "error" is the standard deviation σ which can be made arbitrariy sma by increasing the number of points N and is a measure of the expected deviation from the mean vaue of a samping process. We aso have shown, by Chebyshev s inequaity, that in the imit of infinite steps N, the samping recovers exacty the desired integra over f, which to compute was the aim of the current chapter. 2.2 Markov-Chain Monte-Caro The direct samping procedures presented in the ast chapter are very easy to understand both from a demonstrative point of view but aso from the mathematica side, as one may remember the ony necessary ingredients were the weak aw of arge numbers, the centra imit theorem and the Chebyshev inequaity. The atter was proven in (2.23) whie the weak aw of arge numbers and the centra imit theorem are the common theorems from stochastics that every physics student is aware of since his or her undergraduate ectures on data anaysis. However, to understand non-direct samping processes, a itte more of stochastics is necessary, which is presented on a minima eve in the next section. Subsequenty, we wi be abe to understand Markov-Chain Monte Caro processes and the necessary conditions for a successfu impementation of these powerfu samping procedures Markov-Chains In this section, we wi give a brief introduction to Markov-Chains and the possibiity to determine an unknown and compex probabiity distribution from a running Markov-Chain. For that, in the contrast to the previous section, we wi restrict ourseves to a configuration space Ω which is countabe (athough it may be infinite). This is not a serious restriction, since if Ω was consisting of a subset of continuous variabes, we just take a fine grid on this subset such that we end up with a countabe number of grid-points to be considered, such that resuting set is countabe. Since we can make the grid arbitrariy sma, we can find a choice of the grid which does not affect the integras that we are interested in, i.e. those given by (2.10). In the foowing, Ω is countabe and X := (X n ) n N is a stochastic process in the space of configurations Ω, i.e. n, X n takes vaues C n Ω, whereas the probabiity P (X n = C n ) = P (X n = C n (X 0,...X n 1 ) = (C 0,..., C n 1 )) of this event aso depends on the vaues that X i, i = 0,...n 1 has taken in Ω. Definition 2.1 (Markov-Chain) X = (X n ) n N is caed Markov-Chain, if the Markov-condition is fufied, i.e. P (X n = C n (X 0,..., X n 1 ) = (C 0,..., C n 1 )) = P (X n = C n X n 1 = C n 1 ) for a n 1 and C 0,..., C n Ω with P ((X 0,..., X n 1 ) = (C 0,..., C n 1 )) > 0. In other words, the stochastic process X = (X n ) n N in every step n ony depends on the previous step n 1 and does not directy depend on the steps before. Definition 2.2 A Markov-Chain X = (X n ) n N is caed homogeneous, if P (X n+1 = C X n = C) = P (X 1 = C X 0 = C) n 1, C, C Ω. The transition matrix P = (p C C) C C Ω is the Ω Ω -matrix of the transition probabiities p C C = P (X 1 = C X 0 = C).

29 2.2. Markov-Chain Monte-Caro 25 I.e. a Markov-Chain is homogeneous, if the transition probabiities are the same for every step n and do not change as X proceeds. Remark (Properties of the transition matrix) The transition matrix from the above definition is a stochastic matrix, i.e. has the properties p C C 0 C, C Ω C Ω p C C = 1 C Ω. The entries are therefore non-negative (since they represent probabiities) and the sum over a compete row equas 1 since in every step n X n takes a vaue in Ω, no matter which vaue it took in the previous step. With these definitions, it is very simpe to find a forma expression for the probabiity distribution of X n for a certain n N. Suppose the probabiity distribution for the variabe X n 1, i.e. P (X n 1 = C) =:, is known for a C Ω. Then it is straight forward to compute µ (n 1) C µ (n) C = P (X n = C) = C Ω P (X n = C X n 1 = C)P (X n 1 = C) = p CC µ (n 1) C = (pµ (n 1) ) C. (2.26) Iterative appication of (2.26) resuts in the forma expression for the probabiity distribution µ (n) C, which reads µ (n) C = (pµ (n 1) ) C = (p n µ (0) ) C, (2.27) where p n is the nth power of the transition matrix p and µ (0) is the initia probabiity distribution of X 0, which is defined independenty of the Markov-Chain. With these definitions and the properties (2.26), (2.27), we have aready found the most genera properties of Markov-Chains. From now on, we wi ony consider homogeneous Markov-Chains and introduce two further fundamenta concepts in the framework of Markov-Chains, namey irreducibiity and stationarity, which wi be fundamenta for the successfu impementation of Markov-Chain samping. Definition 2.3 (Irreducibiity) Let C, C Ω be configurations. The configuration C Ω can be reached from C Ω, if This is denoted by C C. n N 0, such that (p n ) C C > 0. C and C, with C, C Ω are said to be communicating, if C C and C C. This is denoted by C C. A homogeneous Markov-Chain is caed irreducibe, if C C, C, C Ω, in other words, if a configurations C, C Ω are communicating. The ast definition is very important for our purpose and therefore we wi formuate irreducibiity once more in a descriptive definition, which is competey equivaent to Def. (2.3). Definition 2.4 (Irreducibiity 2) A homogeneous Marko-Chain is caed irreducibe, if for any configuration C Ω any other configuration C Ω can be reached with probabiity arger than zero in a finite number of steps.

30 26 2. Monte-Caro Methods in a Nutshe In a physics context, the ast definition is commony and mistakeny, referred to as ergodicity, which is however not the same as irreducibiity. Athough in the context of a Markov-Chain samping they are usuay fufied at the same time, ergodicity is stricty a requirement on the stationary distribution of a Markov-Chain, whie irreducibiity is a property of the transition probabiities of a Markov-Chain, as we see from Def. (2.3). In addition to irreducibiity, an important concept that we wi now introduce is the concept of a stationary distribution. Definition 2.5 (Stationary distribution) A probabiity distribution π on Ω is caed stationary distribution with respect to p (the homogeneous Markov-Chain), if π(c) = C Ω π( C)p C C. (2.28) As we see from (2.28), a stationary distribution, once reached, wi never be changed in the subsequent steps of the Markov-Chain, i.e. n N, with µ n = π µ m+n = π, m N. Definition 2.6 (Reversibiity) Let π be a probabiity distribution on Ω. A Markov-Chain is caed reversibe with respect to π, if π(c)p C C = π( C)p CC, C, C Ω. (2.29) If a Markov-Chain is reversibe with respect to π, then π is a stationary distribution of this Markov-Chain, as we see from summing over both side of the above equaity π(c) = C Ω π(c)p C C = C Ω π( C)p CC, where in the first equaity we have used the properties of a transition matrix p as expained in remark (2.2.1). In order to obtain the fundamenta resut of this section, the Markov-Chain convergence theorem, we need one ast definition. Definition 2.7 (Aperiodicity) A configuration C Ω is caed aperiodic, if and ony if n N, such that m N, (p m+n ) CC > 0. A Markov-Chain is caed aperiodic if a configurations are aperiodic with respect to its transition matrix p. Unti now, we have ony introduced concepts from the theory of Markov-Chains and one may ask, what we gain from those. The answer wi be given by the foowing two theorems, that wi not be proven in this thesis as their proofs require the understanding of many additiona concepts of stochastic processes and stochastic anaysis that we do not want to introduce at this point. Instead, we refer the interested reader to mathematica books on stochastic processes, such as [75, 41]. Theorem (Uniqueness of the stationary distribution) An irreducibe Markov-Chain has at most one stationary distribution. If a stationary distribution is found, it is unique.

31 2.2. Markov-Chain Monte-Caro 27 Theorem (Markov-Chain convergence theorem) Let (X n ) n N be an irreducibe, aperiodic Markov-Chain, with a stationary distribution π. We define m C (N) N as the number of times the Markov-Chain has passed the configuration C Ω after N steps in tota. Then, we find m C (N) im = π(c). (2.30) N N This resut is independent of the starting distribution µ (0). The Markov-Chain convergence theorem is a fundamenta theorem of stochastic anaysis and the cornerstone for a advanced Monte-Caro methods that are not based on direct samping. To understand this we shorty review the concepts we have introduced so far. In section 2.1, we posed the quantity we wish to find, which was given by the integra I(w, f) = Ω dc f C Ω dc w = dc π(c)g C, (2.31) C Ω where g C = f C /w C and we have repaced p C = w C / dc w C by π(c). We expect that we have no chance to compute the probabiities π(c) sufficienty exact without a gigantic effort in programming and computer power, so we ask what can we do to obtain the correct probabiities? The Markov-Chain convergence theorem tes the answer: buid a Markov-Chain, which is as simpe as possibe but fufis the requirements of the theorem and which has the probabiity distribution π(c) as its stationary distribution. Then after running this Markov-Chain for a sufficienty arge number of steps, we wi obtain the probabiity distribution within arbitrary sma error. That is exacty what we are going to do in the foowing section The Metropois agorithm The aim of this section is to set up a Markov-chain, which is abe, through a very simpe sampe procedure, to determine the probabiity distribution π(c). The probabiities are formay determined by the expression w C π(c) =, (2.32) Ω dc w C where we have access to w C for any configuration C Ω but because of its compexity, can not compute the integra in the denominator. A Markov-Chain is determined by its starting distribution µ (0), which was unimportant for the convergence theorem and its transition matrix p. To successfuy obtain the probabiity distribution π after a Markov-Chain samping, we have to ensure that π is the stationary distribution with respect to p, and that the Markov-Chain (and therefore p) is irreducibe and aperiodic. Aperiodicity can usuay be fufied easiy by setting up a Markov-Chain, which has the property p CC > 0, i.e. a non-zero probabiity to stay in a certain configuration. This aready ensures aperiodicity of p as one reaizes immediatey from its definition (2.7). To ensure that the Markov-Chain has the correct stationary distribution, we use the Metropois procedure, invented by N. Metropois et a in their famous 1953 paper [132]. This agorithm uses the fact that a reversibe (see Def. (2.6)) Markov-Chain with respect to the probabiity distribution π, has the same probabiity distribution as its unique stationary distribution. Since the probabiities are formay known to us, we insert them into (2.29) to get w C w p Ω dc w C C C = C p Ω dc w C CC (2.33) or by bringing both transition probabiities on the same side of the equation, we end up with p CC p C C = w C w C (Detaied Baance). (2.34)

32 28 2. Monte-Caro Methods in a Nutshe This ast equation is the ceebrated detaied baance condition, which is the simpest but aso most powerfu procedure to obtain correct transition probabiities p CC, with the stationary distribution πc = w C / Ω dc w C. Athough the space Ω may be very huge, the detaied baance condition can be fufied very simpy because it connects ony the two states C and C, making no additiona restriction on the residua transition probabiities or configurations. To set up the Metropois procedure (remember that we have to impement this as a numerica procedure), we use the idea of acceptance-rejection samping. The transition probabiities are decomposed in a product of two probabiities, namey the proposa and acceptance probabiity, i.e. p C C = p acc pprop. The proposa C C C C probabiities can be assigned arbitrary with the ony restriction that they must behave as probabiities, i.e. 0 p prop 1 and C C C Ω p prop C C = 1, C, C Ω. (2.35) To iustrate the concept of proposa and acceptance probabiities, suppose we are at a certain step of the Markov-Chain in the configuration C, that we now propose a new configuration C for the next step and the probabiity of this proposa is given by the proposa probabiity p prop. As an exampe, et us consider C an infinitey extended grid, with every grid-point having 4 neighboring grid-points and the grid-points represent possibe configurations. In a particuar step of the Markov-Chain, we sit on a certain grid-point, which we ca C. For the next step of the Markov-Chain, we have to propose a possibe configuration C, which we do by aowing ony the neighboring grid-points to be reached in a singe step of the Markov- Chain but a with the same probabiity 1/4. I.e. we simpy pick any neighboring point of C and ca this the configuration C. This proposa has nothing to do with the actua probabiities of the configurations, it may even be that some of these neighboring grid-points C have the actua probabiity π(c) = 0 but this is irreevant in the proposa of these configurations. After the proposa of a new configuration for the next step, this configuration can be accepted, with probabiity p acc or can be rejected with probabiity 1 pacc. If it is accepted, C becomes the configuration C C C C for the next step in the Markov-Chain. On the other hand, if it is rejected, the od configuration C stays as the configuration for the next step in the Markov-Chain. To bring these proposa and acceptance probabiities into the detaied baance equation, we just rewrite (2.34) with these newy introduced probabiities, which then reads p acc CC pprop CC p acc pprop C C C C = w C w C pacc CC p acc C C C = w Cp prop C C. (2.36) w Cp prop CC The right side of the right equation is fixed from outside, both by the weights w C, w C, which are determined by the probem itsef, and by the proposa probabiities p prop CC, pprop, that we have introduced (and C C consider for the moment, to be arbitrary). The Metropois procedure (or Metropois agorithm) now proposes the foowing: Suppose the Markov-Chain is in the nth step in configuration C, now pick a new configuration C from the set of aowed configurations (this set of course depends on C) with probabiity p prop C C. This configuration is accepted with the Metropois probabiity p acc = min {1, w } Cp prop C C. (2.37) C C w Cp prop CC The configuration C is either accepted, then it is the configuration of the (n + 1)th step or rejected, then the od configuration C stays and aso becomes the configuration of the (n + 1)th step. Using this agorithm, we have aready ensured, that πc = w C / dc w C is the stationary distribution of this Markov-Chain. At this point, it is important to reaize that the stationary distribution of the Markov- Chain does, in no way, depend on how compicated (or simpe) we chose for the proposa probabiities p prop. As soon as the acceptance probabiities are chosen according to equation (2.37), the stationary distribution of the Markov-Chain stays invariant under the change of the proposa probabiities. This

33 2.2. Markov-Chain Monte-Caro 29 important fact gives us the freedom to chose the proposa probabiities such that the ast condition of the Markov-Chain convergence theorem is fufied, which is the irreducibiity condition, as defined in Def. (2.3). The irreducibiity condition, can on a forma eve, be fufied by choosing a Markov-Chain that makes π ergodic, which is defined as Definition 2.8 (Ergodicity) A stationary distribution π of a Markov-Chain is caed ergodic, if its support ies in an irreducibe component of the Markov-Chain. The support of a probabiity distribution π is supp(π) := {C Ω π(c) > 0}. It is cear, if we had chosen a Markov-Chain such that π is ergodic, then the above choice of the acceptance probabiities woud make the Markov-Chain irreducibe. In physics, irreducibiity is often meant to be equivaent to ergodicity and since we understand that in the context of the Metropois agorithm both are equivaent, we wi from now on aways tak about ergodicity. As mentioned before, there is no genera recipe to ensure ergodicity in an arbitrary Markov-Chain and therefore this has to be done ad hoc, depending on the distribution π one is interested in. In genera, ensuring ergodicity is the most difficut part in setting up a Markov-Chain samping method because, on the one hand there is no recipe how to actuay do that and on the other hand, ergodicity in a mathematica sense is often not sufficient. To iustrate that, we use the foowing exampe. Suppose configuration space Ω consists of two distinct irreducibe subspaces, i.e. Ω = Ω 1 Ω 2 with Ω 1 Ω 2 = and there exists ony a singe configuration C Ω 1 from which it is possibe to reach Ω 2 during a Markov process. This setup wi be ergodic in a mathematica sense, since it wi be possibe to reach any configuration from any other one in a finite number of steps. In a numerica simuation, if Ω 1 is very arge, it wi never happen with finite probabiity, that a Markov-Chain starting in Ω 1 wi ever reach a configuration in Ω 2 and therefore the numerica simuation wi with finite probabiity give incrediby wrong resuts. Of course, this exampe is an extreme case and no one, famiiar with MCMC methods, wi set up such a Markov-Chain. It simpy iustrates that athough a Markov-Chain might ook ergodic, it might turn out to be not ergodic enough for a numerica simuation. One understands the concept even more ceary, when reaizing that the configuration space Ω might be so compex that it is impossibe to overview it and the ony predictions that one can make are oca ones, i.e. on sma subsets of Ω which may not even be connected in some manner. Mosty, the ony way to verify that a Metropois agorithm is working, is to perform numerica simuations and test if it is working correcty by benchmarking the resuts with other anaytica and computationa methods, which however might ony be possibe for some imiting cases, where are methods are appicabe 5. Suppose now we have impemented the Metropois agorithm correcty and have ensured ergodicity. Then the Markov-Chain convergence theorem appies and in a very ong Metropois samping procedure, we wi be abe to determine the probabiity distribution π. But how does this hep us to compute the integra I(f, w) = Ω dc f c Ω dc w = dc π(c)g C (2.38) c Ω required in this simuation? Of course, it woud be possibe to appy a direct samping procedure now, since π is known. But this woud be inefficient since we woud have to perform a samping procedure twice and we aso woud have to store the vaues of π for every configuration C in Ω. Aso it woud be necessary to determine π for every singe configuration C in the Markov-Chain samping which woud take a tremendous amount of computation time. The idea is to use the Markov-Chain convergence theorem, as we wi see. We start with setting up a Markov-Chain as it is described by the Metropois agorithm. Then we et this Markov- Chain evove during a numerica samping procedure but, instead of counting the number of times m C (N) every configuration C Ω has been reached during the samping, we simpy add up the vaues g C. For every step n in the Markov-Chain, we know the configuration of this step C, therefore we can compute g C = f C /w C and just add this to a variabe which we ca J (n) (w, f). This means that in every step ony two variabes have to be determined, first J (n) (w, f) = J (n 1) (w, f) + g C and second the step number n (which is just a running integer). After n steps of the Markov-Chain J (n) (w, f) can be expressed as J (n) (w, f) = C Ω m C (n)g C. (2.39) 5 As for instance exact diagonaization is ony appicabe to finite systems but not in the thermodynamic imit, whereas a huge cass of Monte-Caro methods is appicabe to both finite and infinite systems

34 30 2. Monte-Caro Methods in a Nutshe In the imit of n, we can therefore appy the Markov-Chain convergence theorem, which causes the fraction J (n) (w, f) m C (n) im = im g C = dc π(c)g C = I(w, f) (2.40) n n n n C Ω to converge to the desired integra I(w, f). For a finite number of steps n <, we have found an approximate soution for the integra I(w, f) by appying Markov-Chain samping. The error, however, is again purey statistica in nature and we again can appy error anaysis toos, known from stochastics. The main difference here, is that in contrast to direct samping, the individua steps of the samping procedure are not independent from each other as it is the case for direct samping processes. The configuration C of a certain step n in the Markov-Chain strongy depends on the configuration C of the previous step n 1 and for this reason can not be independent. This means that the error estimation formuas, obtained for the direct samping have to be modified to cover Markov-Chains. As a concusion of this section, we present some remarks. Remarks: In the Markov-Chain samping, a that has to be cacuated expicity are the weights w C and the functions f C for certain configurations C which are passed by the Markov-Chain (the configuration space Ω contains much more points than actuay wi be passed during a simuation). The computation and subsequent accumuation of g C sha in the foowing be caed a Monte-Caro measurement. The most difficut part in a Markov-Chain samping is, as mentioned above, the determination of the proposa probabiities and the possibe configurations for the current step to take. This has to be done very carefuy and it has to be verified afterwards via simuation if the resuting Markov-Chain is irreducibe. Often it is computationay demanding to determine the functions f C (much more than w C ), which depends on the physica probem that has to be soved. Since the subsequent steps in the Markov- Chain samping depend very strongy on each other, not much information is gained by adding g C in every step and it is often usefu to have certain intervas during the samping process where nothing is summed up and the Markov-Chain just evoves. Athough the starting configuration of the Markov-Chain µ (0) was not important for the Markov- Chain convergence theorem, it is cear that it wi have an impact on the convergence speed of a Markov-Chain and aso on the vaues of g C at the very beginning of the samping process. It is therefore often usefu to wait a significant number of steps before starting to accumuate g C. This is often caed equiibration of the Markov-Chain. The formaism that we used here does not distinguish between cassica and quantum mechanica configurations C. In fact, the ony difference between cassica and quantum Monte-Caro (QMC) is the fundamenta difference in the configuration space but nothing wi change in the samping procedure, as we wi see in the next chapter. Metropois samping is one reaization of Markov-Chain samping, however it is the most weknown and powerfu reaization. The Metropois samping is the basis for a kinds of Monte- Caro processes, which are mosty either direct impementations of the Metropois agorithm or extensions with the same basic idea. A the Monte-Caro processes that are used during this thesis are based on Metropois samping, the specific choice of configuration space and the corresponding proposa probabiities is what makes them unique and why they are not simpy caed Metropois or Markov-Chain Monte-Caro Statistica Data Anaysis for Markov-Chain Samping Procedures In the direct samping procedures, every measurement (i.e. every randomy determined vaue of g C ) was independent from a other measurements, since the bare probabiities were used for the samping. This

35 2.2. Markov-Chain Monte-Caro 31 was used in equation (2.18) to express the variance of the sum of the variabes in terms of the variance of one singe variabe. For the Markov-Chain samping, this is impossibe, since the singe measurements are not independent from each other. This can again be best understood by the exampe of a Markov-Chain on a grid. Suppose for every grid-point C, a different vaue g C wi be accumuated during the samping. In a singe step of the Markov-Chain, we can move from a certain grid-point to one of its neighbors. This means the measurement of the (n + 1)th step depends on the nth step because it can ony be made on a neighbor of the nth configuration. Therefore, a ot of steps m are necessary in the Markov-Chain, unti a configuration C at step n + m is independent of the configuration C at step n. This number of steps is usuay unknown and in a compicated samping may aso strongy depend on the configurations and vary throughout the samping procedure, which usuay wi make it impossibe to estimate the number of intermediate, not measured steps m unti the next measurement wi be independent from the previous one. For the Markov-Chain samping therefore one has to use sighty more advanced error anaysis toos than for the direct samping, which we wi discuss now. Again, the statistica error of the accumuated variabe F N is expressed through its variance Var(F N ), where we use the same notation as in the previous data anaysis section. F N is the vaue accumuated through a Markov-Chain samping with N measurements during the sampe, i.e. F N = 1 N N f i, (2.41) i=1 where the f i are the individua measurements. Of course, a singe samping procedure to obtain one vaue F N takes a ot of time and computer power and therefore repeating this procedure to get a precise variance of F N is not an option. One has to find error anaysis toos, which can be appied on a singe run and sti give the precise variance of F N. To do this, we again express the variance of F N through the singe measurements ( Var(F N ) = FN 2 F N 2 = 1 N 2 ( ) N N f i ) 2 f i 2. (2.42) The expectation vaue... is inear in its individua arguments and the f i are a equay distributed, i.e. f i = f j f and Var(f i ) =Var(f j ) Var(f) for a 1 i, j N. The difference to the previous data anaysis section is that f i f j f i f j since they are not independent. This eads to Var(F N ) = 1 N 2 = Var(f) N N f i f j f i f j = 1 i,j=1 N 2 ( i j 1 + ( f ) if j f 2 ) N Var(f) i=1 i=1 N Var(f) + f i f j f i f j i j = Var(f) N (1 + 2τ A), (2.43) where we have defined the auto-correation time τ A as i j τ A = ( f if j f 2 ). (2.44) 2N Var(f) The auto-correation time contains N(N 1) summands, where it is impossibe to make any statement on their behavior during a certain samping procedure. The best resut for the auto-correation time is that of independent variabes, when τ A = 0 vanishes, on the other hand, for variabes which are not independent, there is no argument that prevents τ A from scaing as τ A = O(N). The atter woud be the worst case scenario, because then the tota variance Var(F N ) woud be constant and the Markov- Chain samping woud not converge to a fixed vaue 6. From the Markov-Chain convergence theorem, we know that a Markov-Chain that is set-up propery (i.e. fufiing the conditions of (2.2.2)) wi converge to a stationary distribution in the imit of infinite steps. From that we directy concude, that a proper 6 It is obvious that a Markov-Chain, which is converging to a stationary distribution is not aowed to have fixed non-zero variance in the imit of infinite steps.

36 32 2. Monte-Caro Methods in a Nutshe Markov-Chain wi have an auto-correation time τ A which does not scae with the number of steps N (at east ess than inear order). In a good samping process, for sufficienty arge N, τ A wi no onger depend on N and the tota standard deviation wi again scae as σ(f N ) = O(N 1/2 ). We can aso formuate the previous argument from another perspective. As we have seen, a finite auto-correation time resuts from non-independent variabes f i, f j. If we sufficienty increase the number of intermediate samping steps m between two subsequent measurements, these variabes wi become independent again and the auto-correation time vanishes but the number of measurements scaes ineary with N, the tota number of steps in the samping. If m is owered now, the variabes are no onger independent and τ A becomes finite but the number of measurements sti scaes ineary with N, which means the variance in tota may be arger but it is not aowed to depend on N. The ony case, when τ A scaes with N is the case when it is impossibe to bring τ A to zero by the increase of m. In this case the measurements wi never become independent, no matter how many intermediate steps ie between two measurements, which can ony be the case when the Markov-Chain does not converge. With the introduction of the auto-correation time, we have found a powerfu too to anayze possibe Markov-Chain samping processes. Suppose we coud determine the auto-correation time τ A in a certain samping process. Then we have a direct measure for the Markov-Chain we have constructed: either τ A remains finite in the imit of infinite steps (i.e. is independent of N for N sufficienty arge), such that the Markov-Chain converges to a stationary distribution, or τ A becomes infinite in this imit, and the Markov- Chain ceary does not converge. The atter means we have made a mistake in the impementation and did not propery ensure ergodicity. If the auto-correation time is finite, we obtain the error of the samping process in terms of the standard deviation 1 + 2τA σ(f N ) = σ(f) N. (2.45) Athough the auto-correation time is a very powerfu quantity, we ony benefit from its introduction, if it is possibe to determine τ A during a certain samping procedure. Finding ways to determine τ A during a samping process is therefore the aim of the remainder of this chapter. Binning Anaysis The most common procedure to obtain the auto-correation time is to perform binning anaysis. Starting with the origina set of measurements f (0) i, with i = 1,..., N, we iterativey obtain a binned set of measurements by averaging over two consecutive entries: f () i = 1 ( f ( 1) 2i 1 2 ) + f ( 1) 2i, with i = 1,..., N N 2. (2.46) These bin averages f () i are ess correated than the origina measurements, since they beong to two distinct (imaginary) measurements with an increased number of intermediate steps m () = 2 m (0), where m (0) is the number of intermediate steps in the origina samping process. On the other hand, the mean vaue of the binned averages is aways the mean of the origina measurements i f (0) i. We can estimate the error of the binned variabes, using the variance formua for independent variabes, which we know to be incorrect but, however, converges to the correct error in the imit where the bins become independent of each other. Using (2.45) with τ A = 0, we obtain σ(f () N ) Var(f () i ) N = 2 N N i=1 ( f () i f () ) 2. (2.47) Suppose after steps of appying the binning, the f () i had been independent from each other. Then in the next step of the binning the variabes f (+1) i woud be aso independent from each other and the two factors of 2 appearing in σ(f (+1) N ) woud cance each other resuting in σ(f (+1) N ) = σ(f () N ). On the other hand, if the f () i had not been independent from each other, σ(f () N ) from Eq. (2.47) woud be the wrong expression for the standard deviation, i.e. woud underestimate the exact expression. Since in the next binning step, the variabes wi become more independent, (2.47) wi be a more reaistic expression,

37 2.3. Diagrammatic Monte-Caro Methods 33 which means that σ(f () () N ) wi grow with increasing unti it saturates at the point, where the f i are competey independent. Therefore the error estimate from (2.47) converges to the correct error estimate σ(f N ) = im σ(f () N ). (2.48) Since in a reaistic samping procedure, the number of avaiabe steps have to be finite, one can adjust the number of tota steps of the samping according to the convergence of σ(f () N ), i.e. unti σ(f () (+1) N ) σ(f N ) = ɛ, (2.49) where ɛ is the aowed residua deviation that has to be determined from outside. This binning anaysis gives a reiabe recipe for estimating errors and autocorreation times. After convergence against σ(f N ) is achieved, the auto-correation time τ A can be obtained by rearranging (2.43) via ( ) 2 σ(f N ) 1 τ A = 1 2 σ(f (0) N ). (2.50) The number of steps in the samping procedure must be chosen at east thus arge, that this convergence is achieved, otherwise no error estimation is possibe and athough one can just use the simpe assumption of independent variabes and obtain the corresponding error, the correction coming from the fact that the variabes are not independent may be severa order of magnitude arger than this error obtained from this estimation. This is for instance iustrated in [4] We concude this section with some remarks. Remarks: The auto-correation time is a very important too, as we have seen in the previous section, to give an estimate on the errors of resuts from a Markov-Chain samping procedure. However, the name auto-correation time may be miseading, at east from our point of view. Commony, τ A is said to give the number m of intermediate steps necessary to ensure the independence of the measured variabes. As we have seen, this number woud actuay be given by 2, where is the number of binning steps unti convergence is observed. This number is usuay much arger than the auto-correation time, which is aso what we observed from our Monte-Caro simuations. There exist severa important error estimation procedures for Monte-Caro methods which we are not going to expain since binning was sufficient for our case. One method to mention is the Jackknife anaysis, expained in [188]. The aim of this method is not to find the auto-correation time but instead to give a reiabe error estimation for variabes that are functions of Monte-Caro resuts and therefore need to be anayzed with respect to error propagation and cross-correations in the case of mutipe variabes. Same impementations of Markov-Chain samping wi have different auto-correation times and it may happen that an agorithm vaidated with one set of parameters wi become non-ergodic when certain parameters are changed. This is for instance the case for spin-systems, when a symmetry broken phase occurs for a given set of parameters. Therefore it is very important to be carefu with the anaysis for these systems and it may happen that for a given physica system, different agorithms have to be used in different parameter regimes. 2.3 Diagrammatic Monte-Caro Methods In this section, a certain cass of Monte-Caro methods is introduced, namey the so-caed Diagrammatic Monte-Caro Methods (DiagMC), which sampe Feynman diagrams of a perturbation expansion of the tota Hamitonian H = H 0 + V to obtain the interacting Green s functions of a quantum-mechanica probem. We wi start with a short overview on cassica Monte-Caro methods and the difference to quantum Monte-Caro (QMC) methods, then introduce the idea of DiagMC with the formaism used throughout this thesis and finish with a brief outook on the famous sign probem arising in QMC appications.

38 34 2. Monte-Caro Methods in a Nutshe From Cassica to Quantum Monte-Caro As mentioned aready in the previous sections, in genera there is no fundamenta difference between cassica and quantum Monte-Caro methods, for both cases, after finding the most suitabe configuration space, the Metropois agorithm is appied as introduced before. The main difference is the choice of the configuration space, which is, of course, different for cassica and quantum mechanica probems (the keyword for QMC is word ines ). This difference can be understood very easiy by the foowing exampe. Consider a system consisting of N interacting partices in a harmonic osciator. We now first ook at a cassica system and then switch to the quantum anaogue. For a cassica probem, the system is described by a cassica Hamitonian function H({x 1,..., x N, p 1,..., p N }), which is a function of the individua coordinates and momenta of the partices. The Hamitonian for N partices in a harmonic osciator, which are interacting by a two-partice potentia U(x i, x j ) is given by H({x 1,..., x N, p 1,..., p N }) = N i=1 p 2 i 2m mω2 x 2 i U(x i, x j ), (2.51) where the N cassica momenta p i, i = 1,..., N and the N cassica coordinates x i, i = 1,..., N are independent variabes. To now determine the average of a given observabe O(x 1,...x N, p 1,..., p N ) which might be a function of a momenta and coordinates (or some subset of those), we have to determine the integra O = 1 d N xd N p O(x 1,...x N, p 1,..., p N ) e βh({x1,...,x N,p 1,...,p N }), (2.52) Z i,j where the partition function Z is defined by the integra Z = d N xd N p e βh(x1,...,x N,p 1,...,p N ) (2.53) and β = 1 k B T is the inverse temperature. The compicated integra from (2.52) can be determined by Markov-Chain Monte-Caro. A set of possibe configurations C and weights w C is directy visibe from the form of (2.52). The possibe configurations are the momenta and coordinates, i.e. C = (x 1,...x N, p 1,...p N ) and the possibe weights are then given by which eads to the observabes w C = e βh C = e βh(x1,...,x N,p 1,...,p N ), (2.54) g C = O(C) = O(x 1,..., x N, p 1,..., p N ). (2.55) With these definitions, we have found everything necessary for a successfu Markov-Chain Monte-Caro samping (remember section 2.2.2) except the proposa probabiities which for instance coud be distributed uniformy (which woud be a bad distribution but we are not interested in the best choice for this exampe). With our knowedge on Markov-Chain Monte-Caro from the ast section, it is cear that with these simpe choices, we have aready created a Monte-Caro method for determining averaged observabes for a cassica interacting many-body probem. What changes when switching to a quantum-mechanica many-body probem in the same setup, i.e. interacting quantum partices (but for simpicity with cassica statistics) in a harmonic osciator? In the Hamitonian a the variabes x 1,...x N, p 1,..., p N are repaced by operators, which can not simpy be expressed by numbers, especiay not at the same time because p and x are not commuting. To obtain an integra expression for an operator average O, which is aso a function of momentum and coordinate operators now, one has to switch to a functiona integra representation, i.e. O(x 1, τ 1,..., x N, τ N ) = 1 Z QM d N xd N p O(x 1, τ 1,..., x N, τ N )e S eff, (2.56) with S eff = β 0 dτ N i=1 m 2 ( xi (τ) τ ) 2 + m 2 ω2 x i (τ) U(x i (τ), x j (τ)). (2.57) i,j

39 2.3. Diagrammatic Monte-Caro Methods 35 The partition function Z QM is defined equivaenty as Z QM = d N xd N p e S eff. (2.58) In order to find an integra expression for an observabe average in the quantum case, it was necessary to switch to functiona integra representation, from which another degree of freedom arises, namey the (imaginary) time τ. Therefore possibe configurations for the quantum-mechanica probem woud be the "cassica" configurations C cass = (x 1,..., x N, p 1,..., p N ), where we ca 1/m xi τ = p i together with the time τ, i.e. C QM = (C cass, τ). For a quantum mechanica probem, the time is an additiona variabe in configuration space, whereas this is not the case for a cassica probem. Therefore one can either think of totay different configurations (as we wi do in the foowing chapters) for a quantum mechanica probem or one can imagine samping so-caed word ines instead of cassica points in configuration space. These word-ines are nothing ese but the cassica variabes as a function of τ on a interva τ [0, β]. Athough we have totay negected the statistics of the partices in this formaism (we have used cassica, distinguishabe partices), the main difference from QMC to cassica MC is cear from this exampe, namey the (imaginary) time τ as an essentia part of configuration space for QMC agorithms. In the foowing chapters of this thesis, we wi not use word-ine samping and our configurations wi be very different from any cassica configurations, however, we wanted to iustrate the main difference between QMC and cassica MC agorithms, which is not a different samping method but rather a different configuration space Ω Genera Formaism of Diagrammatic Monte-Caro Methods For a quantum-mechanica many-body probem, there exist severa possibiities to obtain physica quantities, beginning by thermodynamic properties determined by the partition function, static operator averages, such as the density distribution, or dynamic quantities, such as the dynamica structure factor or correation functions. The most desirabe quantities one woud determine are the n-body Green s functions of the system, from which every other static or dynamic observabe can be determined. One way to determine the Green s functions of a quantum system, is to use diagrammatic Monte-Caro methods, which can be formuated in both equiibrium and non-equiibrium frameworks. In this section, we wi give a brief introduction to the idea of DiagMC methods and the formuation of those with enough detai to describe the ater impementation of the continuous time impurity sovers. DiagMC methods are diversey used throughout the scientific community and our aim is not to describe them in great detai or review the most recent deveopments. However, we woud ike to emphasize that DiagMC methods, paired with very advanced resummation techniques, have recenty regained a great interest in condensed matter physics and are on the way (together with arge-size custer DMFT methods) to sove exacty theoretica modes such as, for instance, the Hubbard mode at haf-fiing (see for instance [111, 149, 146]). The aim of DiagMC methods is to determine the Green s functions G αα (τ) of a system described by a Hamitonian H = H 0 + V, where H 0 can be soved exacty (anayticay or numericay) and V is an operator that can not easiy be diagonaized simutaneousy with H 0. The Green s function G αα (τ) is then determined by G αα (τ) = 1 ( ) Z Tr T e βh c α (τ)c α (0), (2.59) where the partition function is Z = Tr ( e βh) (2.60) and the time-ordering operator T orders fermionic operators according to their imaginary time, i.e. T A(τ)B(τ ) = θ(τ τ )A(τ)B(τ ) θ(τ τ)b(τ )A(τ) (2.61) for some arbitrary, fermionic operators A, B. In the Heisenberg representation, time-dependence of operators means time-dependence with respect to the tota Hamitonian H, in other words A(τ) = e τh Ae τh. (2.62)

40 36 2. Monte-Caro Methods in a Nutshe Equation (2.59) is usuay not directy evauabe, since H 0 and V are not simutaneousy diagonaizabe. Therefore one has to find another expression for the Green s functions. The exponentia in (2.59) can be reexpressed using the identities with the definition of e βh = e βh0 e βh0 e βh = e βh0 S(β), (2.63) The operator S can be determined by anayzing the Cauchy probem S(τ) = e τh0 e τh. (2.64) τ S(τ) = e τh0 (H 0 H)e βh = e τh0 V e τh0 S(τ) V (τ)s(τ), S(0) = 1, (2.65) which is formay soved by integration S(β) = 1 β 0 dτ V (τ)s(τ). (2.66) In equations (2.65) and (2.66), the time-dependence of V (τ) is no onger with respect to H but to H 0, this is commony referred to as the Dirac representation. In the foowing, we wi no onger use the Heisenberg representation. The action of the time-ordering operator T on other operators is the same in both representations. Iteration of the above equation (2.66) then eads to the Neumann series β τ1 τn 1 S(β) = 1 + ( 1) n dτ 1 dτ 2... dτ n V (τ 1 )...V (τ n ), (2.67) n=1 where it is important to keep the ordering of the operators V (τ) as in the above equation (τ i τ i+m with m N). If one of the operators is in the incorrect order, for instance if τ i+1 τ i, this beongs to the evoution in negative time direction β 0 and can be corrected by an additiona minus sign. It is therefore usefu to aso appy the time-ordering operator in this context and rewrite β τ1 τn 1 S(β) = 1 + T ( 1) n dτ 1 dτ 2... dτ n V (τ 1 )...V (τ n ), (2.68) n=1 such that artificia time-ordering is no onger necessary since the time-ordering operator introduces minus signs whenever they are necessary (i.e. when operators are ordered incorrecty). With the timeordering operator present in (2.68) it is aso no onger necessary to keep the bounds of the integration as in (2.68), and instead a of the imits for the integras become 0 to β with the incusion of the combinatoria factor n! that corrects for overcounting. The resut is the Dyson series ( 1) n β β β S(β) = 1 + T dτ 1 dτ 2... dτ n V (τ 1 )...V (τ n ) = T e β 0 dτ V (τ). (2.69) n! n=1 The ast equaity is a forma reexpression of the Dyson series, but it must be emphasized that this is ony shorthand for the whoe series since it is impossibe to express the exponentia without using the infinite Dyson series (the exceptions are ony the trivia cases when V and H 0 are simutaneousy diagonaized). With this reformuation, the Green s function can be expressed as the soution of the foowing probem: where ( Tr e βh0 T e ) β 0 dτ V (τ ) c α(τ)c α (0) G αα (τ) = ( Tr e βh0 T e ) = β 0 dτ V (τ )... 0 = Tr ( e βh0... ) T e β 0 dτ V (τ ) c α(τ)c α (0) T e β 0 dτ V (τ ) 0 0, (2.70) denotes the average with respect to H 0. Equation (2.70) is aready of very simiar form as (2.31), which was the starting point for the Metropois agorithm and took the form Ω G MC = dc w Cg C Ω dc w. (2.71) C

41 2.3. Diagrammatic Monte-Caro Methods 37 We must now find a possibe set of configurations and weights to impement a Monte-Caro samping computing the Green s functions. A possibe way to do this is to express the partition function through the infinite Dyson series again. This eads to Z = T e β 0 dτv (τ) 0 = ( 1) n n=0 n! β 0 β dτ 1... dτ n T 0 n V (τ ) =1 0 Ω dc w C, (2.72) where we have identified the configurations C and the respective weights w C according to n C = (n, τ 1,..., τ n ), and dcw C = ( 1)n T V (τ )dτ. (2.73) n! The factor n V (τ ) =1 0 in the definition of the weights above wi usuay be very difficut to determine and ooking from a diagrammatic point of view, one immediatey reaizes that this expectation vaue is aready the sum over a Feynman diagrams for a given perturbation order n. Therefore it is aso possibe, starting from the definition of the configurations and weights (2.73) to construct new configurations and weights by decomposition or resummation of od weights. For instance, suppose V is of quartic order in fermionic operators, then for a given perturbation order n, there exist 2n! abeed Feynman diagrams. A possibe choice for configurations and weights then coud be C = (n, k, τ 1,..., τ n ), and dcw C = ( 1)n n! =1 (nth order diagram #k) 0 n dτ, (2.74) where 1 k 2n! abes the individua Feynman diagrams at perturbation order n. The ast exampe can be seen as the starting point for any diagrammatic Monte-Caro method. Commony, it is not the bare abeed diagrams that are samped but instead resummation techniques are appied, for instance it is possibe to resum a diagrams of a given structure, ending up with either bod-ined diagrams 7 or diagrams with renormaized interactions. As aready mentioned, the possibe reaizations for diagrammatic Monte- Caro methods is very diverse and for our purpose it is not necessary to go into further detais. We wi aways use configurations and weights as defined in (2.73). As pointed out above, this can be understood as a diagrammatic Monte-Caro method, where the resummation is done such that a diagrams of a given perturbation order are contained in the weight w C. The ast missing ingredient for the impementation of the Metropois agorithm (except the proposa probabiities which have again to be designed for every probem individuay) are the g C s. We write the Green s functions as G αα (τ) = from which it is cear that for C = (n, τ 1,...τ n ) g C,αα (τ) = ( 1) n n! T c α(τ)c α (0) n =1 V (τ )dτ 0 ( 1) n n! T n =1 V (τ )dτ 0 = =1 Ω dc w Cg C,αα (τ) Ω dc w, (2.75) C T c α(τ)c α (0) n =1 V (τ ) 0 T n =1 V (τ. (2.76) ) 0 The numerica vaue of g C,αα (τ) can be obtained on two distinct ways, either via a direct cacuation, as we wi do it in the ater chapters or again by using diagrammatics. For the atter, one simpy has to remember that w c incudes a diagrams resuting from nth order perturbation theory and therefore g C,αα (τ) contains a (n + 1)th order diagrams from a diagrammatic expression of the Green s function G αα (τ) (connected and disconnected). With the definition of the g C s, the weights w C and the configurations C, we have done the preface for the impementation of the continuous-time quantum Monte-Caro sovers that we expain in the foowing sections. We concude this section with a brief introduction of the sign probem, which usuay arises in quantum Monte-Caro processes. 7 In our exampe this woud mean that in the samping process, the interacting Green s functions are aready incuded instead of the non-interacting ones.

42 38 2. Monte-Caro Methods in a Nutshe The Sign Probem in Monte-Caro Processes From the definition of possibe weights in the previous section, i.e. w C being diagrams. It becomes cear that some of the w C may become negative when deaing with fermions, since fermionic diagrams are not stricty positive in contrast to the bosonic case. Therefore π(c) is negative, which excudes it from being a probabiity and no Markov-Chain can be created by using the weights. It is possibe to circumvent this probem by reaizing that w C = w C sign(w C ) and rewriting the expression for the Green s function samping as Ω G αα (τ) = dcw Cg C,αα (τ) Ω Ω dcw = dc w C sign(w C )g C,αα (τ) C Ω dc w Ω dc w C C Ω dc w C sign(w C ) = G αα(τ) sign, (2.77) where we have defined the positive weighted Green s function Ω G αα (τ) dc w C sign(w C )g C,αα (τ) Ω dc w (2.78) C and the average sign Ω sign dc w C sign(w C ) Ω dc w. (2.79) C Both the average sign and the positive weighted Green s function can be determined in a singe samping procedure according to the positive weights w C without any approximation. At first gance, this might sove the probem of negative weights w C competey and avoiding negative weights in Monte-Caro woud never be an issue. However, the important fact that makes Monte-Caro superior to many other methods is that the Monte-Caro error is under contro. Now suppose the samping error (N) of G αα (τ) was obtained in a Monte-Caro simuation with N subsequent MC measurements. This error can be expressed as (N) = 0 N, (2.80) where the prefactor 0 is unimportant for the moment. Now, the error (N) of the Green s function resuting from the same Monte-Caro samping woud, according to (2.77), be given by (N) = (N) sign = 0 sign 2 N. (2.81) This error is the error beonging to a direct Monte-Caro samping of the Green s function with a reduced number of steps Ñ = sign 2 N. To obtain an error comparabe to N Monte-Caro steps in systems N with positive weights, one has to perform an increased number of steps in a system that shows sign 2 aso negative weights (i.e. when sign < 1). This means, athough negative signs are handed by (2.77) without restrictions on a forma eve, it may be that the number of necessary Monte-Caro steps to obtain an acceptabe error is increasing up to an intractabe number, which makes Monte-Caro simuations very inefficient or even impossibe. To understand how severe this increase of necessary Monte-Caro steps can become for fermionic systems, we take a short ook at the average sign (2.79) for a DiagMC samping as introduced in the previous section. Then Ω sign = dcw C Ω dc w C = Z = e βv (f F f B ), (2.82) Z B where Z B is the partition function of a corresponding (fictive) bosonic system, β is the inverse temperature and V is the voume of the system and f F, f B are the free energies per voume of the fermionic, bosonic system, respectivey. From (2.82), it is cear that the average sign of a system decreased exponentiay with the system size and the inverse temperature. This fact means that simpe, straightforwardy impemented quantum simuations of arger fermionic systems at sufficienty deep temperatures become impossibe and is the imiting probem for exact quantum simuations of interacting fermion systems on the basis of Monte-Caro processes. Very often, this is referred to as the sign probem of QMC methods. However, recenty deveoped DiagMC methods (aso caed bod-ine diagrammatic Monte-Caro methods, BodDiagMC) have found a perfect way to make use of the sign probem [149, 150]. Athough the idea is rather simpe, the impementation therefore is, however, very demanding:

43 2.3. Diagrammatic Monte-Caro Methods 39 Imagine the configuration space Ω consisting of the bare diagrams, as in the exampe from the previous section. Now, construct new configurations C i = {C i1,...c in } consisting of severa od configurations, such that w Ci = n =1 w C i 0 for most of the new configurations C i. Perform a Monte-Caro samping with the few residua, non-zero weights w Ci, where the sign probem is much ess severe, resuting from the effectivey decreased system size. On the first view, this procedure seems very artificia. However, it turns out that this kind of Monte- Caro samping can be achieved by appying resummation techniques, aready known from diagrammatic treatments of condensed matter systems and that these methods are indeed quite powerfu. With this brief introduction of the sign probem, we concude the more genera part on Monte-Caro and diagrammatic Monte-Caro methods and come to the derivation of the so-caed continuous-time quantum Monte-Caro methods, which wi be used to sove the Anderson impurity mode from section (3.1).

44 40 2. Monte-Caro Methods in a Nutshe

45 3. The Anderson Impurity Mode and Continuous-Time Monte-Caro Methods In this chapter, we wi introduce the Anderson Impurity mode (AIM), a famous theoretica mode describing two interacting fermionic partices couped to a non-interacting bath. The AIM was first introduce to describe the Kondo-effect in soid state systems and its great importance was even increased after one reaized that the same mode is strongy connected to the DMFT approximation, as we wi see in the foowing sections. To successfuy appy the DMFT equations, it is essentia in every iteration to cacuate the correation functions of an AIM and use these in the iterative DMFT procedure. Therefore, we are interested in theoretica (computationa) methods to determine those correation functions. In this chapter, we wi discuss two such methods. First, the exact diagonaization of a quantum system with ony few degrees of freedom and, second, Continuous-Time Quantum Monte-Caro (CT-QMC) methods which are abe to determine exact correation functions for arbitrary bath sizes without any additiona computationa cost. This ast point is very important, since, as we wi see, reducing the bath to a finite number of quantum states decreases the spectra resoution of the DMFT approximation dramaticay. 3.1 The Anderson Impurity Mode Our interest in the Anderson Impurity mode (sometimes aso caed singe impurity Anderson mode, SIAM) resuts from the fact that we have to determine the corresponding correation functions in order to use them during the DMFT iterations. Depending on the physics that are investigated (i.e. repusive interactions, attractive interactions, superfuidity or spin orbit couping), the Hamitonian of the AIM that we are considering sighty changes but the genera structure wi aways stay the same. The AIM is a theoretica mode describing two interacting fermionic partices (which we wi abe with spin indices σ =, ), both couped to a non-interacting bath. It is usuay expressed through the Hamitonian H AIM = H oc + H hyb + H bath, (3.1) consisting of the oca Hamitonian H oc = σ µ σ n σ + Un n, (3.2) the Hamitonian of the non-interacting bath, with the bath operators creation and annihiation operators d σα and d σα, H bath = ɛ σ,α d σαd σα (3.3) σ,α

46 42 3. The Anderson Impurity Mode and Continuous-Time Monte-Caro Methods and the hybridization with the bath H hyb = σ,α V σ,α c σd σα + h.c.. (3.4) The bath energies ɛ σ,α are rea parameters, whie the hybridization parameters V σ,α may be compex. The chemica potentia µ σ is aowed to be different for the different components σ =, in order to describe popuation imbaance. As for the attice probem, we woud ike to find an effective action S eff, which ony contains the two components σ =, such that a states α have been integrated out, so that the interacting Green s functions of the ast two components can be written as G σ (τ) = 1 Z eff σ D[c σ, c σ ]c σ(τ)c σe Seff. (3.5) In contrast to the previous chapter, the expression of these Green s functions via an effective action can be formuated by an exact transformation and the exact effective action can be determined anayticay from the parameters {V σ,α, ɛ σ,α }. To prove this, we start with the compete action the AIM, S AIM, S AIM = S oc + S hyb + S bath, (3.6) where we again have separated this into a oca part, the pure bath and the hybridization between oca part and bath S oc = dτ c σ(τ) ( τ µ σ ) c σ (τ) + U dτn (τ)n (τ), (3.7) σ S bath = dτ σ,α d σα(τ) ( τ + ɛ σ,α ) d σα (τ) (3.8) and S bath = dτ σ,α V σ,α c σ(τ)d σα (τ) + h.c.. (3.9) To diagonaize this expression in the time domain (i.e. to get rid of the derivative τ ) and simpify the foowing steps, we switch to Matsubara frequency representation, where the singe parts of the action read S oc = n= c σ(iω n ) (iω n µ σ ) c σ (iω n ) + U c (iω n)c (iω n)c (iω n)c (iω n + iω n iω ), σ n,n,n (3.10) S bath = d σα(iω n ) (iω n + ɛ σ,α ) d σα (iω n ), (3.11) S bath = n= σ,α n= σ,α V σ,α c σ(iω n )d σα (iω n ) + h.c.. (3.12) To integrate out the d σα, d σα degrees of freedom, we make use of the integration aw for Grassmann variabes, which states D[ξ, ξ]e λµ ξ λ M λµξ µ+ λ ( η λξ λ +ξ λ η λ) = (det M) e λµ η λ(m 1 ) λµ η µ. (3.13) By defining the indices λ (σ, α, iω n ), the matrix M σ,α,iωn = iω n ɛ σ,α and the Grassmann variabes = Vσαc σ (iω n ), we can integrate out the bath degrees of freedom and find η σ,α,iωn ( ) D[d ασ, d ασ ]e SAIM = (iω n + ɛ σ,α ) α,σ σ,α,n ( ) e S0+ σ,α,n c σ (iωn) Vσα 2 c iω+ɛσ,α σ (iω n). (3.14)

47 3.1. The Anderson Impurity Mode 43 The prefactor of this expression resuts from the determinant in (3.13) but doesn t pay any roe (except when one is interested in the thermodynamics of the AIM) since it can be absorbed into Z eff in (3.5). In the ast step, we wi find the hybridization function of the AIM Γ σ (iω n ) = α V σα 2 iω n + ɛ σα, (3.15) with which we can again define a Weiss function G σ by Gσ 1 (iω n ) = iω n µ σ Γ σ (iω n ) = iω n µ σ α V σα 2 iω n + ɛ σα. (3.16) Here, the Weiss function has no further index than σ since it is not yet associated with any attice site of the system as it was the case in the previous chapter. Inserting the Weiss function into the effective action eads to S eff = dτdτ c σ(τ)gσ 1 (τ τ )c τ + U dτn (τ)n (τ), (3.17) σ which ooks identica to the effective impurity action of the DMFT equations (1.58). We have found two representations of the AIM effective action, from which we can obtain the correation functions with respect to the c σ, c σ degrees of freedom. First, the action defined by (3.6), which is oca in time but has additiona degrees of freedom of the bath that we are not interested in, and second, the effective action defined by (3.17), which has no additiona degrees of freedom but is non-oca in time due to the non-oca hybridization functions Γ σ (τ). In the foowing sections, we wi work with both representations, depending on the computationa method that we are interested in. However, one shoud note that the effective action coming from the DMFT equation (i.e. that which we wish to sove) has the representation (3.17) and can therefore ony be fuy captured by (3.6) if the bath is infinitey arge 1. The residua (but nevertheess main part) of this chapter, we wi derive computationa methods to sove the AIM mode or, more precisey, cacuating the correation functions, which appears rather simpe but sti demands for very advanced methods to be soved exacty. This difficuty simpy arises from the non-ocaity in time of the hybridization function and must not be circumvented by any approximation (otherwise DMFT woud simpify to a static Mean-Fied theory) Exact Diagonaization Exact Diagonaization (ED) is a very simpe, yet often used approximation to sove the AIM presented in the previous section. As we have seen, one representation of the AIM is that of an interacting, two eve system, couping to an infinite bath. If the bath was finite (which we take as about 4 6 bath orbitas), one possibiity to sove the AIM woud be to numericay impement the Hamitonian as a matrix (in a given compete basis of states) and diagonaize it numericay. Here, the size of the matrix (the number of states) scaes exponentiay with the number of orbitas, which makes ED impossibe for arger systems. On the other hand, DMFT produces a Weiss function G, from which a the parameters {V σα, ɛ σα } woud have to be determined. This woud aso be an impossibe task for a arge system 2 and therefore it is ony possibe to use ED as an approximate method, where the approximation itsef comes from the restriction to a finite bath. When restricting onesef to a finite system, the expicit numbers for the bath energies and hybridization parameters are obtained from a east squares fit to the DMFT Weiss function and subsequent diagonaization of the resuting matrix. In this thesis, ED is never used as an impurity sover for DMFT iterations and therefore, we wi not go further into the detais of this method. However, we used ED to benchmark the Monte-Caro methods described in the foowing sections and numericay impemented during the work on this thesis. For this benchmarking, we reverse the probem by choosing some arbitrary parameters {V σα, ɛ σα }, and then determining the Weiss function G (which in this direction is exacty possibe by (3.16)) and comparing Monte-Caro and ED resuts. In this case, no approximation is made and both resuts are exact and shoud be equivaent an so we are abe to verify the Monte-Caro agorithm. 1 The Weiss function is a bounded function on a finite interva, i.e. τ [ β, β], which means it beongs to a separabe but infinite dimensiona space, which can ony be fuy reproduced by taking into account an infinite number of bath orbitas. 2 V 2 Neither the space of Green s functions nor the basis functions posses a mathematica structure that aows for a systematic ɛ iω determination of the parameters {V σα, ɛ σα}.

48 44 3. The Anderson Impurity Mode and Continuous-Time Monte-Caro Methods 3.2 Continuous-Time Auxiiary Fied Quantum Monte-Caro Agorithm In this section, we derive the continuous-time auxiiary fied quantum Monte-Caro agorithm (CT-AUX), as an impurity sover for the Anderson impurity mode, based on Monte-Caro samping and an auxiiary fied decomposition of the interaction part of the AIM. Continuous-time sovers, based on an auxiiary fied method were first deveoped by Rombouts et a. [154, 155] in 1998 for sma Hubbard attices. Monte- Caro sovers on the Basis of a Trotter decomposition, the so-caed Hirsch-Fye agorithms [86], have been used much earier to treat a sma number of magnetic impurities in metas. The formuation of CT-AUX that we present in this section has been deveoped by Emanue Gu et a. in 2008 [65] and is presented in detai in [65, 67, 70]. We wi mainy foow the origina derivation, i.e. [65], whie we present some of the cacuations in more detai. In particuar, we present an exact derivation of the weight formuas based on functiona integration, whereas the derivation in the origina pubications is based on approximative formuas, deveoped for the Hirsch-Fye agorithm. The resuting weight formuas are, however, identica. The CT-AUX impurity sovers have been used aso in custer DMFT approximations [190, 111], where a custer of mutipe sites N = 4, 8, 32,... has been used as the impurity. In this approaches, the sefenergy becomes momentum-dependent and the resuts become exact with increasing custer size. The CT-AUX agorithm is perfecty suited for custer probems, because a custer impementation is straightforward, as we wi see at the end of this section. However, our intention is to use CT-AUX as an impurity sover for the RDMFT approximation, for which it is usefu because it is an exact method to sove the impurity probem and it can be extended to a more genera cass of systems, incuding spin-orbit couping or superfuid pairing, as we wi see in the succeeding chapters Configurations, Weights and Measurement Factors In this part, we wi identify the configurations C for the CT-AUX agorithm and derive anaytic formuas for the weights w C and measurement factors g C by appying functiona integration and matrix agebra toos to the AIM. The aim of the CT-AUX impurity sover is to compute the Green s functions of the AIM, described by the Hamitonian (3.1) H = σ µ σ n σ + Un n + σ,α ɛ σ,α d σαd σα + σ,α V σ,α c σd σα + h.c., (3.18) with the creation and annihiation operators for the bath states d σα, d σα and the impurity states c σ, c σ, with σ =,. The corresponding functiona integra representation of the AIM, with the bath degrees of freedom integrated out, is determined by the effective action (3.6), (3.17) S eff = dτdτ c σ(τ)gσ 1 (τ τ )c τ + U dτn (τ)n (τ). (3.19) σ We wi start the formuation of the CT-AUX agorithm in operator representation and ater switch to functiona integra representation, which eads to more genera resuts and aows for an exact cacuation of the weights w C. Foowing the derivation presented in section (2.3.2), we decompose the Hamitonian H into an exacty sovabe part H 0 and the additiona part V, i.e. H = H 0 +V. The specific choice of V determines the CT- QMC method one has to use to sove the AIM. The idea of the CT-AUX sover is to chose V such that V has a positive spectrum. If this is the case, the weights w C wi become positive for any configuration C as can be seen from (2.73) and there wi be no sign probem occurring during the MC samping process 3. The simpest choice of V wi be ( V = U n n n ) + n K 2 β, (3.20) 3 To be precise, when the operator V is chosen such that it possesses a positive spectrum, it is possibe to find a Hubbard- Stratonovich transformation, such that a weights are positive [154].

49 3.2. Continuous-Time Auxiiary Fied Quantum Monte-Caro Agorithm 45 where the roe of the positive parameter K wi become cear soon. The choice of V in (3.20) resuts from the fact that the operator part is zero for zero or doube occupancy of the impurity and takes the vaue U/2 for a singy occupied impurity. The resuting Hamitonian H 0 = H V (3.21) is quadratic in fermionic operators and eads to sovabe Gaussian integras in a functiona integra formuation. To further proceed, we appy a discrete Hubbard-Stratonovich transformation to V, which can be found for any bounded operator with a finite number of different eigenvaues (see for instance [142]) and was in the context of CT-QMC sover first introduced by Rombouts [155] ( V = U n n n ) + n + K 2 β = K e γs(n n ), (3.22) β where the positive parameter γ is defined by cosh(γ) = 1 + Uβ 2K γ = n 1 + Uβ 2K + s=±1 ( 1 + Uβ 2K ) 2 1. (3.23) Figure 3.1: Schematic iustration of the discrete Hubbard-Stratonovich transformation. The four point vertex on the eft side is repaced by the sum of vertices on the right. Equaity in (3.22) can be checked by etting the eft and right side of the equation act on the basis states B = { 0,,, } of the impurity, which are the eigenstates of the eft and right operators with the same corresponding eigenvaues, which proves identity since B is a compete basis. To further simpify the resuting expression for the weight factors, we perform a transformation according to e γsσnσ = e γsσ c σ σ +c σ c σ = e γsσ (e γsσ 1) c σ c σ. (3.24) In this equation, we have introduced the spin variabe σ to be used as a number. Whenever σ takes the pace of a variabe, it is identified as σ +1 for and σ = 1 for. The reason for bringing the creation operators in front of the annihiation operators, is to bring them into correct order for a switch to functiona integra representation, where at equa times c σ is considered to at infinitesima smaer time than c σ. We can now again rewrite the operator V as V = K β e γs(n n ) = K β s=±1 s=±1 σ=, [ e γsσ (e γsσ 1) c σ c ] σ. (3.25) In the DiagMC section, using a perturbation expansion in V, the weights w C were found to be dcw C = 1 n T V (τ )dτ, (3.26) n! =1 where n was the current perturbation order and part of the configuration C = (n, τ 1,..., τ n ). One has to keep in mind, that for this expression the expectation vaue incuding the time ordering operator T means that a permutations of the n distinct times τ i, i = 1,...n must be averaged and mutipied with the corresponding sign ( 1 or +1 depending on the ordering). In our version of the agorithm, we wi ony pick one reaization of these permutations, which wi be the time ordered one. Therefore, the factor n! in the denominator can be dropped and by using (3.25), the weights become n [ dcw C = e γs σ (e γsσ 1) c σ (τ )c σ(τ ) ] ( ) n Kdτ, (3.27) β =1 s =±1 σ=, 0

50 46 3. The Anderson Impurity Mode and Continuous-Time Monte-Caro Methods where we have, for convenience, skipped the index for the infinitesima times dτ. We can reaize two facts, eading to a further simpification of the above expression. First, for every time τ, = 1,..., n, there exists a corresponding cassica spin s = ±1 over which is summed in (3.27). To remove the sum over the cassica spins from the weights, we enarge the configuration C to be [C = (n, τ 1, s 1,..., τ n, s n )] which means the summation over the cassica spins becomes part of the samping process and has no onger to be performed to obtain the weight factors w C. Second, when ooking at the Hamitonian H 0, we see it can be decomposed into a sum of three commuting operators, namey H 0 = K β + σ H σ 0, (3.28) where H0 σ contains ony quadratic operators with index σ and therefore commutes with H0 σ. The weights are therefore a product of the individua weights of the spin components σ =,, eading to dcw C = σ=, n [ e γs σ (e γsσ 1) c σ (τ )c σ(τ ) ] σ =1 0 ( ) n Kdτ e K, (3.29) β with the factor e K coming from the first term in (3.28) and... σ 0 denoting the average with respect to H σ 0. Now, we switch to functiona integra representation to compute the operator averages in the weight factors 4. With this, the average in (3.29) transforms to n [ e γs σ (e γsσ 1) c σ (τ )c σ(τ ) ] σ = D[c σ, c σ ]e Sσ 0eff =1 where the non-interacting, effective action for spin σ reads 0 n [ e γs σ (e γsσ 1) c σ (τ )c σ(τ ) ], =1 (3.30) S σ 0eff = β 0 and the non-interacting impurity Green s function is dτdτ c σ(τ) ( (G σ ) 1 (τ τ ) ) c σ (τ ), (3.31) (G σ ) 1 (τ) = δτ( τ µ σ + U 2 ) Γ σ(τ). (3.32) It is important to note that, the factor U/2 has been added here, which is vaid ony for the CT-AUX agorithm and is usuay not present in the definition of the non-interacting impurity Green s function. This factor comes from the subtraction of U/2n σ in the definition of V in (3.22). The interacting Green s functions obtained by the samping procedure again wi coincide with the origina impurity Green s functions. We wi now switch to a more genera notation to compute the weight factors for the most genera case, e.g. aso directy appicabe to generaized Grassmann variabes ike in Nambu notation or to custer formuations, as we wi see ater. To do so, we introduce the matrices S, A, B with their eements S ij = (G 1 ) i,j (τ i τ j ), A ij = δ ij e γsiσi, B ij = δ ij (1 e γsiσi ), (3.33) 4 It woud aso be possibe to obtain expression for the weight factors by appying Wick s theorem or using approximate formuas as in [65] or a other iterature. However, using functiona integras is simpy the most convenient way for us to dea with manypartice probems

51 3.2. Continuous-Time Auxiiary Fied Quantum Monte-Caro Agorithm 47 where the indices i, j have to be understood as genera indices, abeing for instance times, spins, attice sites and so forth, and the Kronecker deta not ony makes A and B diagona but aso sha be zero for unoccupied quantum states and times 5. Then the integra to be computed is the Grassmann integra I = D[ζ, ζ]e ζ i Sijζj (A A B ζ ζ ) A ) = ( = det(a) D[ζ, ζ]e ζ i Sijζj (1 + ζ B ζ ) = ( D[ζ, ζ]e ζ i (Sij Bij)ζj = det(a) det(s B) A ) D[ζ, ζ]e ζ i Sijζj e ζ B ζ = det(a) det(1 BS 1 ) det(s) (3.34) The infinite matrix BS 1 consists of exacty n non-zero rows, where n is the perturbation order, eading to 1 BS 1 being mainy diagona with the exception of the n competey fied rows. Now we interchange the coumns of 1 BS 1 such that these n coumns are the first n of the resuting matrix and the resuting determinant has to mutipied by ( 1) m, where m is the number of necessary permutations perform this interchange of coumns. To get rid of this factor, we perform the same number of permutations to the rows of the resuting matrix, such that the prefactor becomes unity. The resuting matrix from these two operations is of the foowing structure ( 1 M = B S ) 1 T, (3.35) 0 1 where B S 1 is the restriction of BS 1 to rows and coumns such that B has non-zero eements B and B, resuting from the interchanges of rows and coumns and 1 is the n by n unity matrix. T is a non-zero matrix for which we do not care and 1 is the infinite dimensiona unity matrix. Therefore, the desired determinant det(1 BS 1 ) = ( 1) 2m det(m) = det( 1 B S 1 ) det 1 = det( 1 B S 1 ) (3.36) is identica to the determinant of the n n matrix 1 B S 1 and we find I = det(s) det(ã Ã B S 1 ). (3.37) The determinant of the infinite dimensiona matrix S is just a constant and is independent of the eements of A and B and therefore the same for a weight factors. Its physica meaning is that of the non-interacting partition function Z 0. With the resut from this short detour, we can come back to the cacuation of the average in (3.30), which we can easiy identify as D[c σ, c σ ]e Sσ 0eff n =1 with the inverse configuration matrix consisting of the diagona matrix [ e γs σ (e γs σ 1) c σ (τ )c σ(τ ) ] = Z 0σ det(n 1 σ ({s i, τ i })), (3.38) N 1 σ ({s i, τ i }) = Ã Ã B S 1 = e Γσ (e Γσ 1)G 0σ (3.39) (e Γσ ) m = (Ã) m = δ m e γs σ (3.40) and the reduced Green s function (G 0σ ) m = ( S 1 ) m = G σ (τ τ m ). (3.41) 5 This means that for perturbation order n, ony n diagona eements of A and B are non-zero and the infinite residua eements are a equa to zero.

52 48 3. The Anderson Impurity Mode and Continuous-Time Monte-Caro Methods Finay, the weights w C for a configuration C = (n, τ 1, s 1,..., τ n, s n ) can be written down as ( ) 2 Kdτ dcw C = e K Z 0σ det(nσ 1 ({s i, τ i })). (3.42) β σ The weights are defined in terms of a inverse matrix N 1 merey by convention. For the weights a determinant has to be computed and the determinant of N 1 is the inverse determinant of N. In contrast, for the measurements, we wi need the matrix N and not N 1 and therefore wi aways compute and store the matrix N instead of N 1. The next step in our derivation of the CT-AUX agorithm wi be the computation of the measurement factors g C, which we wi again perform in functiona integra representation. According to (2.76), the measurement factors are obtained by the integras T c α(τ)c α (0) n =1 V (τ ) 0 g C,αα (τ) = T n =1 V (τ. (3.43) ) 0 Since the AIM is diagona in the spins σ, we can again separatey determine spin up and spin down, whie the interchanging term wi be zero. This eads to the measurement factors in functiona integra representation D[c g C,σ (τ) = σ, c σ ]e Sσ 0eff cσ (τ)c σ(0) n [ =1 e γs σ (e γsσ 1) c σ (τ )c σ(τ ) ] D[c [ ]. (3.44) n σ, c σ ]e Sσ 0eff =1 e γs σ (e γs σ 1) c σ (τ )c σ(τ ) We now switch again to a simpified expression using the matrices S, A, B defined above. Equation (3.44) corresponds to the integra J = = D[ζ, ζ]e ζ i Sijζj (A A B ζ ζ )ζ xζy D[ζ, ζ]e ζ i Sijζj (A A B ζ ζ ) 1 det(s) det(ã Ã B S 1 ) det(ã) D[ζ, ζ]e ζ i (Sij Bij)ζj ζ x ζy = (S B) 1 x,y, (3.45) where we performed the same transformations as in (3.34) and simpe integration rues for Gaussian integras of Grassmann variabes. The aim is now to reate the matrix eement in (3.45) to an expression that we can dea with. Therefore, we start rearranging the matrix (S B) using the fact that S as we as (1 BS 1 ) are invertibe 6, whie B is not. We find (S B) 1 = S 1 (1 BS 1 ) 1 = S 1 + S 1 [ (1 BS 1 ) 1 1 ] = S 1 + S 1 (1 BS 1 ) 1 [ 1 (1 BS 1 ) ] = S 1 + S 1 (1 BS 1 ) 1 BS 1, (3.46) which can be further simpified by ooking at the structure of (1 BS 1 ) and B. From the previous cacuations we know, when working in the permuted basis, that (1 BS 1 ) = ( 1 B S 1 T 0 1 ) and B = ( B ) (3.47) and therefore the matrix product (1 BS 1 ) 1 B = ( ( 1 B S 1 ) 1 X 0 1 ) ( B ) = ( ( 1 B S 1 ) 1 B ) (3.48) 6 The matrix S is invertibe since its inverse is the Green s function, and the matrix (1 BS 1 ) is invertibe as ong as the weights are arger than zero, whie configurations with zero weight are simpy not considered.

53 3.2. Continuous-Time Auxiiary Fied Quantum Monte-Caro Agorithm 49 Figure 3.2: Iustration of the four distinct diagrams forming the weight factor w C for a configuration C = (n = 2, s 1 =, τ 1, s 2 =, τ2) consisting of a cassica up- and down-spin. Since the interaction is ony of density-density type, there are ony (n!) 2 = 4 instead of (2n)! = 24 diagrams contributing to w C. vanishes competey on the subspace where no configurations have been inserted. On the subspace with non-zero matrix eements of B, B and aso A are invertibe and we can insert the matrix à into this equation, eading to ( (1 BS 1 ) 1 ( à B = à B S 1 ) 1 à B ) ( ) 0 Nσ ({s = i, τ i })(e Γσ 1) 0. (3.49) This can be transated back to the configuration measurements g C directy, eading to g Cσ (τ x τ y ) = (S B) 1 x,y = G σ (τ x τ y ) + G σ (τ x τ i ) [ N σ ({s i, τ i })(e Γσ 1) ] i,j G σ(τ j τ y ). (3.50) With this, we have derived the formuas for the weights w C and the measurement factors g C and found the proper configurations C = (n, s 1, τ 1,..., s n, τ n ), which are the necessary ingredients for a Markov-Chain samping. The ony required informations are the non-interacting Green s functions G σ of the impurity and the computation of the parameters e γ and e γ at the beginning of the samping process The Samping Procedure For the samping procedure, the proposa probabiities have to be found such that ergodicity (see chapter Markov-Chains) of the Markov-Chain is guaranteed. Since the configurations C = (n, s 1, τ 1,..., s n, τ n ) consist of the perturbation order n and n tupes (τ i, s i ) [0, β] { 1, 1} one possibe choice of changing a configuration C = (n, s 1, τ 1,..., s n, τ n ) woud be to either increase the perturbation order by one to n + 1 by inserting a randomy chosen tupe (τ n+1, s n+1 ) or to decrease the perturbation order by one to n 1 by removing a existing tupe (τ i, s i ), i = 1,...n. These wi, in fact, be the updates that we chose for the Metropois agorithm. The probabiities wi be caed the remova probabiity for decreasing the perturbation order n + 1 n and the insertion probabiity for the increase of the perturbation order n n + 1. The corresponding probabiities are decomposed into a proposa probabiity and an acceptance probabiity, i.e. p(n n ± 1) = p acc (n n ± 1)p prop (n n ± 1). For an insertion of a tupe (τ, s), a random time is picked from the interva [0, β] with uniform probabiity distribution the corresponding probabiity is p 1 = dτ 7 β. The spin s is independenty and uniformy chosen from the set { 1, 1} with probabiity p 2 = 1/2. The proposa probabiity for inserting a tupe (τ, s) therefore becomes p prop (n n+1) = p 1 p 2 = dτ 2β and is independent of n. On the other hand, when removing a tupe (τ i, s i ) from the configuration C = (n + 1,...), this tupe must be chosen to be part of the existing n + 1. This is again chosen uniformy, eading to p prop (n+1 n) = 1 n+1. This choice of proposa probabiities, aowing 7 Practicay, it is impossibe to chose τ in a continuous interva for instance because every random number generator ony retains discrete vaues and dτ has to be understood as the width of the singe intervas from which τ is chosen.

54 50 3. The Anderson Impurity Mode and Continuous-Time Monte-Caro Methods for insertion and remova of pairs (s, τ) seems to fufi ergodicity, since any configuration C = (n,...) can be reached from any other configuration C = (n,...) in n + n steps with finite probabiity by removing a n tupes and inserting n distinct ones (of course, many other possibe ways exist to reach C from C ). With the weight factors from the previous section (3.42) ( ) n Kdτ dcw C = e K Z 0σ det(nσ 1 (C)), (3.51) β σ the acceptance probabiities can be obtained via the detaied baance condition (2.34) eading to the ratio p acc (n n + 1) p acc (n + 1 n) = 2β w n+1, (3.52) (n + 1)dτ w n p acc (n n + 1) p acc (n + 1 n) = K det(nσ 1 (n + 1)) n + 1 det(n 1, (3.53) σ σ (n)) where we have used a shorthand notation and repaced C, C by n, n + 1, respectivey. The acceptance Figure 3.3: Schematic picture of four possibe subsequent steps in the Markov-Chain of a CT-AUX samping. a) starts with two distinct auxiiary spins at distinct times, in the next move, the spin-up vertex is removed (b) and another spin-down vertex is inserted (c). In the foowing step, a spin-up vertex is inserted again but at a distinct time (d). probabiities are then determined according to the Metropois formua (2.37) as { } K det(n p acc σ 1 (n + 1)) (n n + 1) = min 1, n + 1 det(n 1, σ σ (n)) (3.54) { } p acc (n + 1 n) = min 1, n + 1 det(nσ 1 (n)) K det(nσ 1. (n + 1)) (3.55) The updates introduced above have been found to ensure ergodicity for a the parameter regimes, the CT-AUX agorithm has been appied to [67]. However, it may usefu to impement aso other possibe moves of the Markov- Chain, which can reduce the auto-correation time and may be much faster that the proposed insertion and remova updates. For instance one possibe move is to fip an existing spin variabe, i.e. going from configuration C = (n, s 1, τ 1,..., s i, τ i,..., s n, τ n ) to configuration C = (n, s 1, τ 1,..., s i, τ i,..., s n, τ n ). Because of the symmetry of the Hamitonian in the cassica spins, these moves shoud aways have an sufficienty arge probabiity and therefore be accepted (and required) very often, which reduces the autocorreation time significanty. The other advantage of these moves is that they are much faster performed than an insertion or remova of a tupe (s, τ). However, the disadvantage is that they ony contribute minimay to the ergodicity, since the subspace that is reachabe with spin-fipping updates is very sma compared to the compete configuration space. There aso exist other possibe Markov-Chain moves for this probem, however the most important are the insertion and remova updates that have been discussed σ

55 3.2. Continuous-Time Auxiiary Fied Quantum Monte-Caro Agorithm 51 in this part. At the end, one can never be competey sure that ergodicity is ensured with these updates for the whoe parameter regime of the AIM and therefore the samping process has aways to be reviewed when starting to investigate a new physica regime with Monte-Caro sovers. Steps of the Samping Process The individua steps of the samping process are iustrated in the CT-AUX fow diagram in Fig Figure 3.4: Monte-Caro fow diagram for the CT-AUX agorithm. Initiaization: At the beginning of the samping process one possibe configuration must be determined (from outside) as the starting configuration to initiaize the samping procedure. For the current configuration C a possibe update has to be chosen, either the insertion or the remova of a tupe (τ, s). Where both have the same probabiity to not infuence the detaied baance formuas above. A new configuration must be proposed, either with a tupe (τ, s) ess or more, depending on the previous step. The acceptance probabiity for this new configuration is determine according to the Metropois probabiities ((3.54) and (3.55)). Either the new configuration is accepted, then the od configuration wi be repaced or the new configuration is rejected, then the od configuration stays unchanged and the new configuration is discarded. A measurement of the observabes, i.e. the g C s, is performed (these may be many observabes). The whoe procedure except the initiaization is repeated.

56 52 3. The Anderson Impurity Mode and Continuous-Time Monte-Caro Methods Of course, this just the most simpe exampe for a running CT-AUX agorithm. In between, there may be impemented additiona updates, such as the spin-fip updates discussed above. Another important point, aready discussed in chapter 2 is whether it makes sense to measure the observabes in every singe step or if the gain in accuracy is much worse than the oss in computation speed. Therefore, Fig. 3.4 iustrates the basic principe of the CT-AUX samping and a reaistic samping woud ook quite different, athough the basic principe remains unchanged. In the previous two sections, we have derived a the necessary formuas and the basic Monte-Caro agorithm to determine the interacting Green s functions of the AIM as it was described above. We wi extend these formuas a custer of impurities, which are couped to each other, and impurities which are no onger diagona in the spin index, either caused by spin-mixing processes or superfuid pairing and show expicit cacuations for these systems in the foowing chapter. In this chapter, before we come to the derivation of a distinct Monte-Caro sover, we wi derive efficient matrix manipuation formuas for the CT-AUX sover, which speed up the cacuations of the matrices N σ and the weights w C incrediby Fast Matrix Manipuations The formuas for the weights and measurements in the previous section strongy depended on the matrices N σ, Nσ 1 respectivey. In this section, we wi derive formuas how to determine the ratio of the weights for a remova or an insertion n (n ± 1) for the case that the matrices N σ (n) are known and show how to determine the new matrices N σ (n±1) after a move has been accepted. The structure of the n n matrix N σ (n) is determined by equation (3.39), stating (N σ (n) ) 1 ij = δ ij e γsiσ (e γsiσ 1)G 0σ (τ i τ j ). (3.56) For the acceptance probabiity for insertion of the tupe (s, τ), the ratio R = is required. The updated matrix (N σ (n+1) ) 1 can then be written as (n+1) det(n σ ) 1 (3.57) det(n σ (n) ) 1 ( (N σ (n+1) ) 1 (n) (N σ ) = 1 Q R S where the n 1 vector Q, the 1 n vector R and the scaar S are defined as with 1 n and The ratio (3.57) can be rewritten as ( (n) N σ 0 R = det 0 1 ), (3.58) Q = (e γs σ 1)G 0σ (τ τ), (3.59) R = (e γsσ 1)G 0σ (τ τ ), (3.60) S = e γsσ (e γsσ 1)G 0σ (0 + ). (3.61) ) ( (n) (N σ ) det 1 Q R S ) = det ( (n) 1 N σ R S Q ) = S RN (n) σ Q. (3.62) For the acceptance probabiity of an insertion, the matrices N σ (n) are required and the vectors R, Q as we as the scaar S have to be determined. Then the probabiity is given by the vector matrix product of (3.62). If the insertion is accepted, the matrix N (n+1) σ ( P Q N σ (n+1) = R S must be computed. Simiar to (N (n+1) σ ) ) 1, we write (3.63)

57 3.2. Continuous-Time Auxiiary Fied Quantum Monte-Caro Agorithm 53 and compute the matrix product ( ) ( ) P Q 1 = N (n+1) (N (n+1) ) 1 (N = (n) ) 1 Q R S R S ( ) (N = (n) ) 1 P + Q R(3) (N (n) ) 1 Q + SQ (1) R P + RS (4) R Q + S S (2). (3.64) Formua 3.64 contains four different equations, stating (1) (N (n) ) 1 Q + SQ = 0 Q = N (n) Q S, (3.65) (2) R Q + S S = 1 S(S RN (n) Q) = 1 S 1 = S RN (n) Q, (3.66) (3) (N (n) ) 1 P + Q R = 1 P = N (n) (1 Q R), (3.67) (4) R P + RS = 0 RN (n) (1 Q R) + RQ = 0 R = RN (n) S. (3.68) Combining these equations, eads to the desired equations which, step by step, determine the matrix N (n+1) : S = (S RN (n) Q) 1, (3.69) Q = N (n) Q S, (3.70) R = SRN (n), (3.71) P = N (n) + Q S 1 R. (3.72) For a remova, the steps to determine the weights and the updated matrix are even simper. Suppose the remova of the th tupe (s, τ ) has been proposed. Without oss of any information, the matrix N (n+1) can be rearranged such that the th and (n + 1)th component interchange position. Then the ratio of the determinants is simpy determined by And the updated matrix N (n) is determined by rearranging (3.73) to R = S RN (n) Q = S 1. (3.73) N (n) = P Q S 1 R, (3.74) where Q, R are simpy the coumn, row vectors corresponding to the index. The weight ratios and update formuas never require the inversion of a matrix or the computation of a determinant, ony matrix vector mutipications and therefore the whoe procedure scaes with n 2 the perturbation order squared, which makes this agorithm very efficient. As we have previousy seen in the measurement factors there is aso ony required a matrix vector mutipication with the matrix N (n) and the non-interacting Green s functions. This simpicity in the computation of a required ingredients for the samping and measurement procedure confirms the choice of the proposa probabiities from the previous section. With this rather technica aspect of the efficient computation of weight ratios and updates, we finish this introductory section on the CT-AUX agorithm. For further deveopments, technica detais and efficiency anaysis, we refer to the iterature, i.e. [70, 65, 69, 64, 67] Benchmarking and Performance Anaysis When impementing a new numerica method, one has to ensure both that the agorithm achieves correct resuts, i.e. by comparing the resuts to other numerica methods 8, and that the agorithm has good performance, i.e. is accurate and fast enough for the desired appication. To anayze the first point, we compared the resuts of the CT-AUX agorithm for the interacting Green s functions of an Anderson impurity mode with finite bath size S = 6 to the exact resuts obtained from diagonaization of the Hamitonian. A set of resuts, together with a Monte-Caro error anaysis is shown in Fig. 3.5, where a good agreement between MC and ED resuts can be found. Note that CT-AUX is an exact method

58 54 3. The Anderson Impurity Mode and Continuous-Time Monte-Caro Methods Figure 3.5: Comparison of Green s functions for the AIM with three bath states for every spin σ for different interaction strengths U at β = 1. The back dashed ines are the Monte-Caro resuts, obtained with the CT-AUX agorithm and the red ines are the exact resuts from diagonaization of the Hamitonian. The error bars are obtained from a converged binning anaysis. For interactions U = 0, 1, 3, the cacuation used 10 6 Monte-Caro steps, for U = 10 we used 10 7 samping steps. Left: Bath energies ɛ σ = (1, 1, 1), hybridization parameters V σ = (1, 1, 1). Right: Bath energies ɛ σ = (3, 2, 1), hybridization parameters V σ = (1, 2, 3). and the error can be made arbitrary sma by increasing the number of Monte-Caro iterations. Another important point is the speed of the agorithm or in other words, the necessary time to obtain acceptabe resuts from numerica cacuations. This has been done by E. Gu et a. [65] for both the CT-AUX and CT-HYB impurity sovers [70] by anayzing the size of the matrices N σ during the CT-AUX sampings (and M σ for CT-HYB). The majority of time during the samping is spent on the manipuations of these matrices, which means the speed of the agorithm mainy depends on how fast these manipuations can be performed and therefore on the size of these matrices. The growth of the matrix size is shown in Fig. 3.6 for CT-AUX, CT-HYB and Hirsch-Fye agorithms, where the atter is not expained in this thesis. However, from our simuations we do not fuy agree with their concusion, since it is not obvious that sma matrix sizes are desirabe. It is cear that the size can be directy connected to the speed of the agorithm, on the other hand, sma matrices mean strong fuctuations in the measurements and therefore much more samping steps are required to obtain the same accuracy than for arger matrices, where the fuctuations are much smaer. Therefore a compromise has to be found between sma matrix sizes with fast updates and arger matrices but more accurate measurements. The average perturbation order n (which is the average size of the matrices) can be infuenced by the parameter K, introduced to the CT-AUX agorithm at the beginning, by the formua [67, 70] n = K βu n n (n + n )/2, (3.75) which means it can be adjusted by a proper choice of K. For sma U < 5β we chose K = 5 and otherwise K = 1, which seemed to be a better choice than keeping n and therefore the matrices as sma as possibe. Further and much more detaied anaysis can for instance be found in [69, 67, 70]. 3.3 Continuous-Time Hybridization Expansion Quantum Monte- Caro Agorithm In this section, the continuous-time hybridization expansion agorithm (CT-HYB) is derived. This agorithm is in some sense compementary to the previousy introduced CT-AUX, since it is not an expansion in the interaction as for the atter, but an expansion in the hybridization function. This agorithm was 8 This may be ony possibe in some imiting cases, when other methods, such as for instance exact diagonaization, can be appied.

59 3.3. Continuous-Time Hybridization Expansion Quantum Monte-Caro Agorithm 55 Figure 3.6: Average perturbation order of the CT-AUX agorithm (bue ine, here denoted as CT-INT) as a function of (eft) βt, where the ratio U/t = 4 is kept fixed and (right) as a function of U/t when βt = 30 is kept fixed. The vaue t enters the AIM through the hybridization function Γ(τ), which corresponds to a 3-dimensiona Hubbard Hamitonian with hopping t. The matrix size increases amost ineary with the factor βu, which can aso be obtained from (3.75). Figure from [70]. deveoped as impurity sover for the AIM [189, 191] to treat parameter regimes with arge interactions much more efficient than it is done for the CT-AUX sover [192]. It is aso perfecty suited for extended impurity mode, consisting of mutipe orbitas not couped via the hybridization function but via the interaction, as it was done in [191, 193]. As for the CT-AUX sover, we wi stick to the basic formuation of the configurations, weights and observabes and derive the genera samping procedure and eave extensions and improvements of this sover to the next chapter Configurations, Weights and Observabes To find the configurations and weights, we foow an opposite way as for the CT-AUX sover, namey, we start with a functiona integra formuation of the AIM and switch back to operator notation after an expansion of the effective action exponentia in the hybridization function. The effective action of the AIM reads (3.6) S eff = = S 0 [ ] dτ c σ(τ)( τ µ σ )c σ (τ) + Un (τ)n (τ) σ dτdτ σ dτdτ σ c σ(τ)γ σ (τ τ )c σ (τ ) c σ(τ)γ σ (τ τ )c σ (τ ) (3.76) where the first part (i.e. S 0 ) is oca in time, i.e. has a direct operator correspondence. The second part, the hybridization, is non-oca in time since it resuts from the integration over the infinite bath. The idea is now, to expand the partition function in terms of the hybridization function Γ. In the expansion,

60 56 3. The Anderson Impurity Mode and Continuous-Time Monte-Caro Methods we denote n σ as the perturbation order of spin σ and n = n + n as the tota perturbation order. The partition function in the functiona integra representation is given by the integra Z = D[c, c ]e S eff = D[c, c ]e S0 1 n σ dτ c σ dτ c σ c n σ! σ(τ c σ )Γ σ (τ c σ τ c σ )c σ (τ c σ ) = n σ=0 1 n!n! D[c, c ]e S0 σ= n σ=0 n σ σ=, σ=1 n σ σ=, σ=1 σ=1 dτ c σ dτ c σ Γ σ (τ c σ τ c σ ) c σ(τ c σ )c σ (τ c σ ). (3.77) The functiona integra on the right side of this equation has a direct operator correspondence and we can rewrite the integra as an time-ordered operator average... 0 with respect to the Hamitonian H 0 = σ µ σ n σ + Un n, (3.78) resuting in the expression 1 Z = n n!n! σ=0 = n σ=0 T 1 (n!n!) 2 n σ σ=, σ=1 n σ σ=, σ=1 In the ast step, the matrix M 1 σ n σ σ=, σ=1 dτ c σ dτ c σ Γ σ (τ c σ τ c σ c σ(τ c σ )c σ (τ c σ ) ({τ c σ ( M 1 σ, τ c σ dτ c σ dτ c σ ({τ c σ 0 ( det ) M 1 σ T ({τ c σ n σ σ=, σ=1 ), τ c σ }) c σ(τ c σ )c σ (τ c σ ). (3.79) }) has been introduced, which contains the eements ), τ c σ }) = Γ σ (τ c σ i ij τ c σ j ), (3.80) which eads to n σ! products instead of a singe one and enforces the additiona factor 1/(n σ!) in equation (3.79). Note that the sign of the determinant exacty cances with the sign that resuts from the timeordering operator and so there is no sign probem. The integras and the two sums over the perturbation order for spin-up and spin-down operators in (3.79) are supposed to be samped by the Monte-Caro process, which identifies the configurations C = (n, n, {(τ c σ, τ c σ σ ) σ = 1,..., n σ, σ = }, consisting of the perturbation orders n, n for spin up, spin down and the 2n σ times (τ c σ, τ c σ σ ) for the respective creation and annihiation operators. In the Monte-Caro procedure, the creation operators wi be integrated in a time-ordered form, as we as the annihiation operators but the ordering of creation operators with respect to annihiation operators wi not be performed. This cances exacty the factors 1/(n σ ) 2 from equation (3.79) above and eads to the formua for the weights dc w C = σ ( (dτ) 2nσ det M 1 σ ({τ c σ ) n σ, τ c σ }) The next step is then to find an anaytic expression for the trace n σ c σ(τ c σ )c σ (τ c σ ) = n, n e βh0 σ=, σ=1 0 n,n =0,1 σ=, σ=1 n σ σ=, σ=1 c σ(τ c σ )c σ (τ c σ ) c σ(τ c σ )c σ (τ c σ 0 ) 0. (3.81) n, n, (3.82)

61 3.3. Continuous-Time Hybridization Expansion Quantum Monte-Caro Agorithm 57 which is rather easy, since for any configuration C, ony of the four summands in the trace wi be nonzero. As mentioned above, the creation operators stay time-ordered with respect to annihiation operators and therefore in the product in (3.82) either a operators for a given spin component are ordered as c σ (τ c σ n σ )c σ(τ c σ n σ ),..., c σ (τ c σ 1 )c σ(τ c σ 1 ) or as c σ(τ c σ n σ )c σ (τ c σ n σ ),..., c σ(τ c σ 1 )c σ(τ c σ 1 ), where the atter introduces a factor of ( 1) because of the changed ordering. A trace of the first product in the space of spin σ, i.e. n σ = {0, 1}, wi have ony one non-zero contribution, namey from the state n σ = 1, whereas in the second product the non-vanishing contribution resuts from the state n σ = 0, which ceary iustrates for the statement above that ony a singe state n, n contributes to the trace (3.82). The configurations C are best understood in terms of a segment picture of the impurity, which we now introduce. The non-zero trace-factor for the product c σ (τ c σ n σ )c σ(τ c σ n σ ),..., c σ (τ c σ 2 )c σ(τ c σ 2 )c σ(τ c σ 1 )c σ(τ c σ 1 ) beongs to the state n σ = 0, i.e. at time τ = 0 the impurity is unoccupied. Then at time τ = τ c σ 1 a spin-σ partice is inserted and stays in the impurity unti it is removed at τ = τ c σ 1 and the impurity is empty again unti at τ c σ 2 a partice is inserted again and so on. These time intervas when the impurity is occupied by a spin up or spin down partice are caed segments and uniquey define the configurations C. When the trace factor beongs to the occupied state n σ = 1, we can interpret the respective product c σ(τ c σ n σ )c σ (τ c σ n σ ),..., c σ(τ c σ 1 )c σ(τ c σ 1 ) as starting with an occupied impurity, i.e. with an occupied interva [0, τ c σ 1 ], and aso end with an occupied impurity, i.e. with the interva [τ c σ n σ, β]. Combining these two intervas, the resut is a singe interva [τ c σ n σ, τ c σ 1 ], which beongs to a segment that winds around and crosses the interva boundaries from β to zero. For such segments, an additiona factor ( 1) has to be introduced, as expained above, because of exchanged ordering. In the segment picture, the trace-factor can be determined in terms of segment properties and is given by Figure 3.7: Iustration of a configuration C in the segment picture. There are two pairs of annihiation and creation operators for spin up and three pairs of annihiation and creation operators for spin down occupying the impurity for different times. The fied squares stand for annihiation operators, whereas the empty squares denote creation operators. Coored ines depict a fied impurity. The prefactors for the trace of this configuration woud be ( 1) since for the down spins one segment crosses the β, 0 ine. The occupation time τσ occ for spin σ woud be the tota ength of a ines with the corresponding coor and the doube occupation time τ doube is the tota width of the vioet bocks, which indicate the time spans when the impurity is douby occupied. n σ σ=, σ=1 c σ(τ c σ )c σ (τ c σ ) 0 = ( 1) σ sσ e µ σ τ occ σ Uτdoube, (3.83) where s σ is the number of segments for spin σ, which cross the 0, β boundary (s σ = 0, 1), τσ occ is the tota ength of a segments for spin σ and τ doube is the tota overap of the occupied segments of the different spins or in other words the tota time in which the impurity is douby occupied. One shoud note that for the case when n σ = 0, i.e. at zero perturbation order we must aso distinguish between a competey fied or a competey empty impurity in order to aways have a singe contribution from the trace above as for a higher perturbation orders. This can be done by decomposing the corresponding Monte-Caro configuration C into two distinct configuration, i.e. a fied impurity and an empty impurity. A typica configuration is iustrated in Fig. 3.7, expaining the trace-factor s ingredients. Up to now, the configurations C and their corresponding weights w C have been identified. The config-

62 58 3. The Anderson Impurity Mode and Continuous-Time Monte-Caro Methods Figure 3.8: Subset of diagrams contributing to the weight w C of a configuration C = (n = 2, n = 3),...). In front of every diagram, there is the corresponding prefactor ( 1) s+d which comes from the determinant (d) and from the trace factor (s) and can be determined by counting the number n R of hybridization ines that are reversed in time, i.e. s + d = n R. urations C = (n, n, S 1,..., S n, S 1,..., S n ) contain the perturbation order n σ for spin σ and the n σ, τ c σ i ) for both spin components. The corresponding weight is then deter- distinct segments Si σ mined by = (τ c σ i dc w C = ( 1) σ sσ e µ σ τ occ σ Uτdoube σ ( (dτ) 2nσ det M 1 σ ({τ c σ ), τ c σ }). (3.84) Note that the prefactor ( 1) sσ in this expression does not cause an expicit sign probem, since it cances with the sign of the determinant det(m σ ) 1, which is positive when no segment crosses β, 0 and becomes negative otherwise. The ast point resuts from the fact that Γ σ (τ) > 0 for τ > 0, and Γ σ ( τ) = Γ σ (β τ), (3.85) which is cear by definition of Γ in (3.15). After the weights and configurations are found, the next step is to determine the observabes g C, which we wi not do directy but instead in a more subte way. An arbitrary Green s function g(τ) can, by using the identity g(τ) = g(β + τ), be re-expressed as g(τ) = dτ dτ (τ τ τ)g(τ τ ), (3.86) where we have introduced the deta-function β (τ τ δ(τ τ τ), for τ > τ } τ) = δ(β + τ τ τ), for τ < τ. (3.87) The Green s function for spin σ is then determined by G σ (τ) = dτ dτ (τ τ τ)g σ (τ τ ) = 1 dτ dτ (τ τ τ) T c σ (τ )c σ(τ ), (3.88) Z which, remembering the effective action in (3.76), can be formuated as G σ (τ) = 1 Z dτ dτ (τ τ Z τ) Γ σ (τ τ ). (3.89)

63 3.3. Continuous-Time Hybridization Expansion Quantum Monte-Caro Agorithm 59 With the definition of g C, it is immediatey cear from this expression, that g σc (τ) = dτ dτ (τ τ τ) 1 w C w C Γ σ (τ τ ) = dτ dτ (τ τ τ) det(m σ ({τ c σ, τ c σ })) det(m 1 σ ({τ c σ Γ σ (τ τ )., τ c σ }) (3.90) Combining the rues for functiona differentiation, i.e. and matrix inversion Γ σ (τ i τ ) Γ σ (τ τ ) = δ(τ i τ τ + τ )δ σ σ, (3.91) M ij = det( M 1 )(ji) det(m 1 ) = 1 det(m 1 ) det(m 1 ) (M 1, (3.92) ) ji where we used the cofactors ( M 1 )(ji) for the matrix inversion, which has a possibe representation as the derivative of the determinant with respect to the corresponding matrix eement (M 1 ) ji, we finay obtain the formua for the observabes: g σc (τ) = (τ c σ i τ c σ j τ)(m σ ({τ c σ, τ c σ })) ji. (3.93) The deta-function in this formua must either be measured on a fine grid in τ-space, i.e. on a discretized time interva [0, β] or, as we wi do, the observabes have to be performed in another representation which is much more suitabe for continuous, smooth functions (see chapter...). The average occupation n σ for the σ-component and the average doube occupancy n n are other observabes, which can be directy measured in the Monte-Caro samping, which becomes cear when taking the derivatives β n σ = 1 Z As above, this eads to the observabes n σc = 1 w C = τ σ occ βw C µ σ β Z and β n n = 1 Z µ σ Z U. (3.94) and (n n ) C = 1 βw C w C U = τ doube β, (3.95) which can be accumuated without any effort during the Monte-Caro samping and are determined more accuratey than the Green s functions since they are static variabes. With a the necessary ingredients, namey the configurations, the weights and the observabes, we can skip and jump to the next section and set up the Markov-Chain and determine the acceptance probabiities for the samping process The Samping Procedure In the previous section, we introduced the configurations C in ( the segment representation, ) where the segments denote pairs of creation and annihiation operators c σ(τ c σ ), c σ (τ c σ ), which occupy the σ- state of the impurity in the time interva [τ c σ, τ c σ ]. Subsequenty, we expressed the Monte-Caro weights and observabes in terms of these segments. In this section, we wi decide the aowed moves of the Markov-Chain samping process, which we choose to be the insertion and remova of segments and the insertion and remova of anti-segments, a together sufficient to obtain ergodicity 9. We start with the discussion of the insertion and remova of segments as possibe updates. According to the Metropois procedure, the transition probabiities p C C for moving from configuration C to configuration C is decomposed into a product p C C = p prop of a proposa probabiity C pacc C C C p prop and an acceptance probabiity pacc, whereas the proposa probabiity wi be defined by the possibe C C C C moves of the Markov-Chain and the acceptance probabiities are then determined by the Metropois rue (ref Metropois).

64 60 3. The Anderson Impurity Mode and Continuous-Time Monte-Caro Methods Figure 3.9: Iustration of four subsequent possibe steps in the Markov-Chain samping. First, in a), the chain starts with a configuration C consisting of two segments for spin-up and three segments for spindown. In the next step of the Markov-Chain, b), a specific spin-down segment is removed. In step c), an anti-segment for the spin-up component is removed, eading to the merging of the two present segments into a singe one, and finay, in d), a distinct spin-up segment is inserted again. Suppose the current configuration C consists of n σ segments, occupying the σ-state of the impurity. Then with probabiity p = 0.5 we pick a spin-component σ = {, } to update. Possibe updates are either to insert a new segment into a free time interva or to remove one of the n σ existing segments from the impurity. For the insertion, which wi be chosen with probabiity p = 0.5, a time τ c σ has to be seected at which the creation operator c σ(τ c σ ) sha be inserted into the impurity. This time is chosen from the interva [0, β] with uniform distributed probabiity, i.e. p = dτ β. After the time τ c σ has been determined, the agorithm has to check if the impurity is empty or occupied at this time. If the impurity is occupied, this move is rejected and the samping then proceeds with the next step, otherwise when the impurity is empty, the ength max of the empty time interva from time τ c σ to the starting time of the next segment is determined. The annihiation operator c σ (τ c σ ) is then inserted at a uniformy chosen time τ c σ in the interva [τ c σ, τ c σ + max ], i.e. with probabiity p = dτ max. The proposa probabiity for the insertion of this segment is then determined by the product of the two probabiities p prop (n σ n σ + 1) = dτ 2 β max. On the other hand, for n σ + 1 segments occupying the impurity, the probabiity to uniformy pick a certain segment to remove is simpy p prop (n σ + 1 n σ ) = 1 n σ+1. With this, the detaied baance condition states p acc (n σ n σ + 1) p acc (n σ + 1 n σ ) (n = pprop σ + 1 n σ ) w nσ+1 p prop (n σ n σ + 1) w nσ = (n σ + 1) e µσδτ occ σ Uδτ doube det(mσ 1 (n σ + 1)) β max det(mσ 1 (n σ )), (3.96) where in the ast step, the formua for the weights (3.84) was inserted and we introduced the quantities δτσ occ and δτ doube which are the change in the occupation time and doube occupation time between the two configurations. Note that we took the absoute vaue of the determinant ratio since we have shown previousy that the prefactors ( 1) s coming from the determinants cance with the prefactors from the trace-factor, see (3.81). The Metropois condition fixes these probabiities to { p acc (n σ n σ + 1) = min 1, (n σ + 1) e µσδτ occ σ Uδτ doube det(m 1 } σ (n σ + 1)) β max det(mσ 1 (n σ )), (3.97) { β p acc max occ (n σ + 1 n σ ) = min 1, (n σ + 1) e µσδτ σ +Uδτ doube det(mσ 1 } (n σ )) det(mσ 1 (n σ + 1)). (3.98) By incuding updates that insert and remove segments, ergodicity has been obtained on a forma eve since any configuration C = (n, n,...) can be reached from any other configuration C = (ñ, ñ,...) 9 Pease note that the insertion and remova of segments aone is not sufficient to obtain ergodicity, i.e. the insertion and remova of antisegments is required for the Markov chain and not a too that simpy improves the samping procedure.

65 3.3. Continuous-Time Hybridization Expansion Quantum Monte-Caro Agorithm 61 in N = n + n + ñ + ñ steps. However, it turns out that the insertion and remova of segments does not guarantee ergodicity in finite samping processes. This fact can aso be conjectured from the fact that a competey empty impurity and a competey fied impurity are not treated on equa footing with these kind of updates as it is very unikey to insert a segment spanning the whoe time interva [0, β]. This can be adjusted by considering a compementary set of updates, namey the insertion and remova of so-caed anti-segments. Whie a segment is nothing but a time interva (τ c σ, τ c σ) for which the impurity is occupied, an anti-segment is just a contrary time interva (τ c σ, τ c σ ) for which the impurity is unoccupied. To express this in the segment picture, the insertion of an anti-segment ( τ c σ, τ c σ ) is nothing but the spitting of a singe segment (τ c σ, τ c σ) into two distinct segments (τ c σ, τ c σ) ( τ c σ, τ c σ), whie the remova of an anti-segment ( τ c σ, τ c σ ) is the recombination of two subsequent segments into a singe one. The update probabiities are then determined by the formuas (3.97) and (3.98) since the anti-segment moves foow competey the same procedure as the segment moves and the weights are identica 10. As for the CT-AUX sover, further updates can aso be impemented in the CT-HYB sover, which ead to a decrease of the auto-correation times but are not required for ergodicity. A set of such updates woud be the possibiity to not ony insert and remove (anti-) segments but aso to aow for shifting segments in time or to shift ony the starting- or end point of a segment in time. These updates are impemented straight-forwardy by using the detaied baance condition and are not derived here. However, in some situations a compete exchange of the segments for the two different components is required, especiay when the system is approaching magneticay ordered phases [67]. Steps of the Samping Process The Monte-Caro fow diagram for the CT-HYB sover is very simiar to the diagram for the CT-AUX sover, Fig The steps are as foows: 1. Initiaization: At the beginning of the samping process one possibe configuration must be predetermined as the starting configuration to initiaize the samping procedure. 2. For the current configuration C a possibe update has to be chosen, either the insertion/remova of a segment (τ c σ, τ c σ) or insertion/remova of an anti-segment (τ c σ, τ c σ ). Where a four of those possibiities have the same probabiity to not infuence the detaied baance formuas above. 3. A new configuration must be proposed, either with one segment (τ c σ, τ c σ) ess or more (or one anti-segment ess or more), depending on the previous step. 4. The acceptance probabiity for this new configuration is determined according to the Metropois probabiities (3.97) and (3.98). 5. Either the new configuration is accepted and the od configuration repaced, or the new configuration is rejected, and the od configuration stays unchanged (i.e. the new configuration is discarded). 6. A measurement of the observabes, i.e. the g C s, is performed (these may be many observabes). 7. Go back to step 2, unti the specified number of iterations has been achieved. Finay, after deriving the configurations, weights and observabes in the previous chapter and setting up the samping procedure, we wi aso compare the CT-HYB sover with exact diagonaization to check whether it is working correcty or not. In Fig. 3.10, we show a comparison between ED and CT-HYB Green s functions for two distinct hybridization functions, determined by the bath energies ɛ σ and hybridization parameters V σ, which are chosen to be the same for both spin components. Together with these resuts, the performance of the CT-HYB agorithm can be discussed in the same way as the CT-AUX sover from Fig. 3.6, which shows the average perturbation order in dependence of βt and βu. Since CT- HYB is an expansion in the hybridization, it does not scae with U but rather with the factor βt, which is 10 In fact they are ony identica after a rearrangement of the matrix M has been performed, since, in contrast to the segment insertion/remova, for the anti-segment insertion/remova the inserted/removed coumns do not beong to the same segments. Athough this is a trivia modification of the matrix M after the insertion/remova of an anti-segment it must not be forgotten or negected since it totay changes the measurements.

66 62 3. The Anderson Impurity Mode and Continuous-Time Monte-Caro Methods Figure 3.10: Comparison of the interacting Green s function g(τ) = G σ (τ) of an AIM with finite number of bath orbitas S = 3 obtained from a CT-HYB agorithm and the exact resuts from diagonaization of the Hamitonian. The accuracy does not depend on the interaction parameter U, since CT-HYB is an expansion in the hybridization. The Green s functions have been obtained for (eft) bath energies ɛ σ = (1, 1, 1) and hybridization parameters V σ = (1, 1, 1) and (right) bath energies ɛ σ = (3, 2, 1) and hybridization parameters V σ = (1, 2, 3). Athough a very sma number of samping steps, N = 10 6, was used, the agreement is aready very good and can be improved by taking the usua number of steps N = 10 8, which turned out to be optima for most simuations that we performed. the Hubbard hopping parameter and indirecty proportiona to the hybridization. In this chapter, we have found the basic formaism for the continuous-time Monte-Caro impurity sovers CT-HYB and CT-AUX, for which we wi discuss severa extensions and improvements in the foowing chapter and which we wi use ater in combination with RDMFT to investigate the properties of fermionic attice modes with artificia gauge fieds.

67 4. Extensions and Improvements to the CT-QMC Methods The continuous-time Monte-Caro impurity sovers introduced in the previous chapter, namey the CT- AUX and the CT-HYB sover, were so far ony discussed for the case of a singe impurity and with spin a good quantum number, i.e. without terms that mix the spins, as they woud occur for systems which show superfuid ordering or spin-changing hopping parameters. In this chapter, we wi discuss both, the extension of the CT-AUX and the CT-HYB sovers to systems with superfuid pairing and spin-changing hopping processes. We wi aso extend the CT-AUX sover to custer impurities, containing more than just a singe attice site. Additionay, improvements of the measurements in the CT-HYB sover are discussed, which aow for a direct measurement of the sef-energy [74] and which perform the measurements in a different basis compared to the previous chapter, i.e. the measurements wi be performed in a poynomia basis instead of the discretized imaginary-time interva [16]. 4.1 CT-AUX on a L-site Custer So far, the impurities that have been considered to be soved by the CT-AUX agorithm ony consisted of two quantum states, i.e. spin-up and spin-down, which were interacting with each other and both couped to a non-interacting bath 1. Now, we wi introduce an additiona degree of freedom to the system, which can be seen as a spatia one. The Hamitonian of the extended system is given by H = i,jσ t ij c iσ c jσ + h.c. + i Un i n i µ(n i + n i ) + iσ c iσ ˆV iσ + h.c. + H bath iσ, (4.1) where the operators ˆV iσ and H bath iσ are defined as ˆV iσ = k V (iσ) k d iσk, V (iσ) k C (4.2) and H bath iσ = k ɛ iσ k d iσk d iσk, ɛiσ k R (4.3) and the index i = 1,..., L abes the L additiona quantum states. For the case of L = 1, this ceary reduces to the AIM discussed in the previous section. For L > 1, the Hamitonian (4.1) describes a system 1 In a functiona integra representation, the bath is equivaent to a hybridization Γ, which is non-oca in time.

68 64 4. Extensions and Improvements to the CT-QMC Methods of L impurities i = 1,..., L, which are couped to a non-interacting bath Hi bath bath +Hi via the hybridization operators ˆV iσ. Additionay the impurities are couped to each other via the hopping t ij which is eft arbitrary for the case of this derivation. Simiar to section 3.2, the Hamitonian is decomposed as H = H 0 + i W i, (4.4) where the operators W i are chosen as W i = U ( n i n i n ) i + n i K 2 β, (4.5) thereby impicity defining H 0 as H 0 = H i W i, (4.6) which eads to H 0 being quadratic in the fermionic operators c iσ, d iσk. From the commutation reation one directy infers the important resut [W i, W j ] = 0, i, j (4.7) [H, W j ] = [H 0, W j ], j. (4.8) Next, we woud ike to express the partition function in terms of time-ordered operators S i (β), foowing the procedure in section Rewriting the exponentia of the Hamitonian (4.1) as defines the operator which soves the Cauchy probem according to e βh = e β(h Wi) e β(h Wi) e βh = e β(h Wi) S i (β) (4.9) S i (τ) = e τ(h Wi) e τh, (4.10) τ S i (τ) = e τ(h Wi) (H W i H)e τh = e τ(h Wi) W i e τ(h Wi) S i (τ) = W i (τ)s i (τ), S(0) = 1. (4.11) Because of (4.8), W i (τ) simpifies to W i (τ) = e τ(h Wi) W i e τ(h Wi) = e τh0 W i e τh0 (4.12) and the forma soution of (4.11) is given by (compare (2.65), (2.66) and (2.69)) S i (β) = T e β 0 dτ Wi(τ). (4.13) The residua exponentia in equation (4.9) can now be rewritten again, eading to with e β(h Wi) = e β(h Wi Wj) e β(h Wi Wj) e β(h Wi) = e β(h Wi Wj) S j (β), (4.14) being determined by the Cauchy probem S j (τ) = e β(h Wi Wj) e β(h Wi) (4.15) τ S j (τ) = e τ(h Wi Wj) (H W j W i H + W i )e τ(h Wi) = W j (τ)s j (τ), S j (0) = 1. (4.16) Again, because of (4.8), W j (τ) = e τ(h Wi Wj) W j e τ(h Wi Wj) = e τh0 W j e τh0, (4.17)

69 4.1. CT-AUX on a L-site Custer 65 which eads to the forma soution of (4.16) being S j (β) = T e β 0 dτ Wj(τ). (4.18) This soution is independent of whether the same step has been performed for W i before or not, since in the time-dependence of W j (τ) ony the quadratic Hamitonian H 0 is present. After performing the above steps L times for a different states i = 1,..., L, the partition function can be expressed as n i=0 Z = e βh = e βh0 L i=1 S i (β), (4.19) with S i (β) defined by (4.13). To dea with S i (β) during a Monte-Caro simuation, we rewrite it in terms of the we-known Neumann series (2.67) β τ i S i (β) = ( 1) n dτ1 i 1 τ i dτ2... i n i 1 dτnw i i (τ1)...w i i (τn i i ). (4.20) This can aso be done for the time-ordered product in equation (4.19), eading to [ L L β τ i ] S i (β) = ( 1) ni dτ1 i 1 τ i dτ i n i dτn i i W i (τ1)...w i i (τn i i ) i=1 i=1 n i=0 β τ1 τn 1 L = ( 1) n ( dτ 1 dτ 2... dτ n Wi (τ # ) ) n i, (4.21) n=0 L{n i}=n where we introduced L{n, which is the sum over a possibe reaizations of the set {n i}=n i} such that i n i = n and the pacehoders τ #, which are the imaginary times that have to be determined by the order in which the operators are introduced in the Monte-Caro procedure 2. For any W i in (2.68), the discrete Hubbard-Stratonovich transformation (3.22) can be performed, introducing n additiona cassica spins s = ±1, = 1,...n, as in section 3.2. The ony difference between here and the corresponding expression for the singe-site impurity (Eq. (3.26) and foowing) is the additiona index i. For that reason, a the computations from the previous sections can be performed anaogousy by repacing the tupes (τ, s ) by the tripes (τ, s, x ), where x = 1,..., L abes the additiona quantum state i. This eads to a change of the configurations from C = (n, τ 1, s 1,..., τ n, s n ) in the previous section to C = (n, τ 1, s 1, x 1,..., τ n, s n, x n ), which eaves the resuts obtained in the functiona integra formaism competey unchanged. The weights w C of the configurations C = (n, τ 1, s 1, x 1,..., τ n, s n, x n ) are determined by the formua with the matrix w C = e LK ( Kdτ β consisting of the diagona matrix ( e Γ σ ) and the Green s function i=1 ) n σz0σ det ( N 1 σ ({τ, s, x }) ) (4.22) N 1 σ ({τ, s, x }) = e Γσ (e Γσ 1)G 0σ (4.23) m = δ me γs σ (4.24) (G 0σ ) m = G σx x m (τ τ m ). (4.25) As one reaizes from (4.22) and (4.25), the ony difference for the custer probem compared to the singe site impurity is that the Green s functions have an additiona dependence on the indices x, x m. The Green s functions are determined by G 1 σx x m (τ) = { δ(τ)( τ µ σ + U/2) Γ σx (τ) for x = x m δ(τ)t x x m for x x m }. (4.26) 2 In the ast equaity of (4.21), it seems that a factor 1/L! is missing because we switched from a sum in which ony the L individua parts i = 1,..., L are time ordered with respect to each other to a fuy time ordered representation. We did not account for this term, since in the Monte-Caro procedure there wi be aways an impicit ordering between distinguishabe states i, j with i = j such that (4.21) with respect to the MC procedure is the correct expression for the product on the eft of (4.21)

70 66 4. Extensions and Improvements to the CT-QMC Methods As for the weights w C, the measurement formuas stay formay unaffected and read g Cσx x m (τ) = G σx x m (τ) + G σx x i (τ τ i ) [ N σ ({τ j, s j, x j })(e Γσ 1) ] i,j G σx jx m (τ j ), (4.27) where g Cσx x m (τ) is the interacting Green s function beonging to configuration C, as in the previous section. With this, the measurement and weight formuas are determined and one can directy start with the samping procedure. The samping is equivaent to the samping procedure of the singe site CT-AUX method, based on insertion and remova of auxiiary spins s i. The ony difference for the custer probem is that a new configuration contains the tripe (τ i, s i, x i ), whereas x i has to be chosen from the set {1,..., L}. Performing this according to a uniform distribution, i.e. with probabiity 1/L, there occurs an additiona factor 1/L in the proposa probabiities, which eads to a modification of the probabiity ratio of equation (3.53), which has to be mutipied by a factor L. These minor changes are aready sufficient to sove custer impurity probems with the CT-AUX method. However, there is one important difference between the singe site and the custer probem, namey the weights w C can become negative in the custer probem, whie they were aways positive for the singe site probem. As pointed out in section 2.3.3, the average sign decreases with the size of the system, which eads to a sign probem, that increases with the custer size. This restricts the custer version of CT-AUX intermediate custer size and interaction strengths, which is aso pointed out in [68, 67]. The custer version of the CT-AUX method introduced above in combination with custer formuations of DMFT have been used successfuy to anayze momentum seective phase transitions in two-dimensiona systems, such as the momentum seective Mott-transition [68, 64] and the pseudogap transition for attractive interactions [121, 122]. For these investigations, the parameters t ij were chosen such that they reproduce a certain dispersion reation in discretized momentum space instead of mimicking the rea-space hopping from attice site to attice site. This eads to the possibiity of addressing severa regions in momentum space that are expected to be sensitive to the specific phase of the system. The custer formuation of CT-AUX was not used to investigate physica systems in the context of this thesis, however, since its formuation and impementation is straightforward after it has been done for singe-site CT-AUX, we briefy introduced this method in this section. 4.2 Spin-Mixing Formuation of CT-AUX For the Anderson impurity mode as considered up to now, the spin σ has been a good quantum number since the interactions did not change the spin and both spin components couped to a distinct set of bath states. It is easy to imagine that this does not cover the fu range of possibe physics for a singe-site impurity mode. Consider for instance a reaization of the Hubbard mode which has additiona hopping terms, incuding a possibe spin fip of a partice hopping from attice site i to the neighboring attice site j. This woud ead to a system without having spin as a good quantum number. As we saw in the previous chapter, the action of the AIM can be considered as the oca effective action of a attice probem, where a attice site but one have been integrated out. Therefore it is immediatey cear that for such a mode the corresponding AIM woud have to contain coupings of the impurity states to bath states with opposite spin as we. In this section, we wi derive the corresponding impurity sover for such systems. The Hamitonian we are considering has the form H = σ µ σ n σ + Un n + σ c σ( ˆV σ + Ŵ σ) + h.c. + σ H bath σ, (4.28) where we have introduced the non-hermitian operators ˆV σ and Ŵ σ, which are defined as ˆV σ = k V σ k d σk (4.29) and Ŵ σ = k W σ k d σk (4.30)

71 4.2. Spin-Mixing Formuation of CT-AUX 67 and the bath Hamitonian H bath = H bath + H bath, with H bath σ = k ɛ σ kd σk d σk. (4.31) The norma hybridization operator ˆV σ introduces a couping of the spin σ to the bath states (σ, k) via the hybridization parameters Vk σ C, whereas the anomaous hybridization operator Ŵ σ introduces a couping of the spin σ to the bath states with opposite spin ( σ, k) via the anomaous hybridization parameters Wk σ C. Switching to a functiona integra representation and integrating out the bath degrees of freedom, the effective impurity action corresponding to the Hamitonian (4.28) is given by ( ( ) ) S eff = dτdτ c (τ), c (τ) Ĝ 1 (τ τ c ) (τ ) c (τ + U dτ n (τ)n (τ), (4.32) ) where the matrix of Green s functions is defined as ( ) ( Ĝ 1 τ µ (τ) = δ(τ) 0 Γ (τ) Γ + (τ) 0 τ µ Γ (τ) Γ (τ) ). (4.33) The hybridization functions are obtained according to the description in section 3.1, equation (3.13), i.e. Γ σσ (iω n ) = k V σ k 2 + W σ k 2 iω n ɛ σ k (4.34) and Γ σ σ (iω n ) = Vk σ(w k σ) iω n ɛ σ + W k σ (V k σ ) k k iω n ɛ σ, (4.35) k fufiing the reation (Γ σ σ (iω n )) = Γ σσ ( iω n ). As in the derivation of the CT-AUX agorithm, we decompose the Hamitonian (4.28) into an exacty sovabe part H 0 and the residua part V, i.e. H = H 0 + V. The choice of the operator V and the subsequent transformations are competey equivaent as described in detai in the CT-AUX section 3.2.1, i.e. equations (3.20)-(3.25), such that the weights are obtained via dcw C = 1 n T V (τ )dτ (4.36) n! =1 and the configurations C stay unchanged, namey are given as C = (n, τ 1, s 1,...τ n, s n ) with the perturbation order n and the n tupes (τ i, s i ), i = 1,..., n. The expicit cacuation of this expectation vaue can be performed according to our derivation of the weight formuas in the CT-AUX section. However, instead of showing the detaied cacuation with sight modifications again, we can derive the weight formuas in a diagrammatic picture. The weight factor as given by (4.36) is the n-th order term of a perturbation expansion in the interaction and therefore it is cear how to express this term in a diagrammatic picture, as we have shown in Fig. 3.2 for a specific configuration of the standard CT-AUX. The singe addition that must be made to this picture, is that the anomaous propagators G (τ) are non-zero and therefore must be considered in the diagrammatic picture. 0 Fig. 4.1 shows additiona diagrams compared to the non-spin-mixing CT-AUX method, contributing to the weight factor (4.36). The weights for a given configurations C = (n, τ 1, s 1,..., τ n, s n ) are obtained via ( ) 2 Kdτ dcw C = e K Z 0 det(n 1 ({s i, τ i })), (4.37) β where Z 0 is the non-interacting partition function, which is no onger simpy the product of spin-up and spin-down partition function, and the 2n 2n matrix N 1 ({s i, τ i }) is defined by N 1 ({s i, τ i }) = e Γ (e Γ 1)Ĝ0, (4.38)

72 68 4. Extensions and Improvements to the CT-QMC Methods Figure 4.1: Iustration of 12 of the (2n)! = 24 distinct diagrams contributing to the weight factor w C at perturbation order n = 2 with finite anomaous Green s functions G σ σ (τ). The spin conserving norma Green s functions are depicted with straight ines whie the spin mixing anomaous Green s functions are drawn as dotted ines. The four diagrams in the vioet region are those four diagrams which do not invove spin-mixing terms and therefore are the ony ones that woud contribute for the standard CT-AUX agorithm. In the green region, are the four diagrams that consist of spin-mixing terms ony. The four diagrams on the right side consist of both spin-mixing and spin-conserving terms, which is aso the case for the 12 remaining diagrams which we have not drawn here. with the n n distinct, 2 2 dimensiona bocks and (Ĝ0) m = (e Γ ) m = δ m ( e γs 0 0 e γs ) ( G (τ τ m ) G (τ τ m ) G (τ τ m ) G (τ τ m ) (4.39) ). (4.40) One immediatey reaizes that for G (τ τ m ) = G (τ τ m ) = 0 the determinant factorizes to a product of determinants for the distinct spin components ony and therefore to the we-known expression from the CT-AUX without spin-mixing terms. In the notation of the 2 2 bocks, as used above, the Monte-Caro observabes can be expressed in an equivaent manner to the measurement formuas in terms of the matrix N and the non-interacting Green s functions, eading to g C (τ) = Ĝ(τ) + Ĝ(τ τ i) [ N({s i, τ i })(e Γ 1) ] i,j Ĝ(τ j), (4.41) where the [ N({s i, τ i })(e Γ 1) ] i,j refers to the 2 2 dimensiona sub-bock i, j of the matrix N({s i, τ i })(e Γ 1) and g C (τ) is the bock of interacting Green s functions ( ) g C (τ) g g C (τ) = C (τ) (4.42) g C (τ) g C (τ) corresponding to the configuration C. Again, these formuas simpify to the CT-AUX measurement formuas for systems without spin-mixing, as soon as the anomaous Green s functions G σ σ are equa to zero. After obtaining the weight and measurement formuas, the samping procedure is totay equivaent to section except that the weights and measurements have to be determined according to (4.37), (4.41), respectivey. Fig. 4.2 shows a comparison of interacting Green s functions obtained via the extended CT-AUX method with Green s functions obtained from exact diagonaization of Hamitonian of the corresponding Anderson impurity mode, i.e. (4.28).

73 4.3. Superfuid Formuation of CT-AUX 69 Figure 4.2: Interacting Green s functions G and G obtained from Monte-Caro samping (dashed ines) and exact diagonaization (red ines) of an Anderson impurity mode, with fixed fiing but different interaction strengths βu. For strong interactions one can identify sma deviations of the MC resut from the exact resut because a figures have been obtained with 10 6 samping steps, which is ceary too few for strong interactions βu = Superfuid Formuation of CT-AUX To propery describe systems with attractive interactions, i.e. with an interaction Hamitonian H = Un n, (4.43) one has to aow for superfuid pairing, i.e. the anomaous Green s functions (τ) = c (τ)c (4.44) can become finite. To aow for a this, the Anderson impurity mode must contain terms which don t conserve of partice number. The AIM we consider for this probem is given by the Hamitonian H = σ µ σ n σ Un n + σ c σ ˆV σ + h.c. + H bath, (4.45) where the hybridization operator ˆV σ is defined as usua ˆV σ = k V σ k d σk (4.46) and the bath Hamitonian contains the symmetry breaking terms H bath = ( ) ( ) ( d k, d ɛ k k k k k ɛ k d k d k ). (4.47) Integrating out the bath degrees of freedom, this eads to an effective impurity action ( ( ) ) S eff = dτdτ c (τ), c (τ) Ĝ 1 (τ τ c ) (τ ) c (τ U dτ n (τ)n (τ), (4.48) ) with the matrix of Green s functions ( ) ( Ĝ 1 τ µ (τ) = δ(τ) 0 Γ (τ) + 0 (τ) 0 τ + µ 0(τ) Γ ( τ) ). (4.49) The hybridization functions can be obtained from the parameters of the impurity mode according to the formua (3.13), eading to Γ σσ (iω) = k V σ k 2 (iω n + σɛ σ ) (iω n ɛ )(iω n + ɛ ) k 2 (4.50)

74 70 4. Extensions and Improvements to the CT-QMC Methods and 0 (iω) = k V σ k 2 k (iω n ɛ )(iω n + ɛ ) k 2. (4.51) In order to determine the configurations, weights and observabes for the samping procedure, we perform a partice-hoe transformation for the spin-down component of the system, i.e. defining new operators c = c } c = c { c, c } = {c, c } = 1 (4.52) ñ = 1 n. (4.53) These fermionic operators can now be inserted into the effective action, eading to ( ( ) ) S eff = dτdτ c (τ), c (τ) G 1 (τ τ c ) (τ ) c (τ + U dτ n (τ)ñ (τ), (4.54) ) which is the effective action for a spin-mixing system with repusive interactions and non-interacting Green s functions, which are determined via ( ) G 1 δ(τ)( τ µ (τ) = U) + Γ (τ) 0 (τ). (4.55) 0(τ) δ(τ)( τ + µ ) Γ ( τ) With these representation, the configurations, weights and observabes can be directy obtained from the previous section about the spin-mixing CT-AUX method. The samping procedure remains the same as for the origina CT-AUX without any off-diagona terms in the hybridization. The superfuid CT- AUX method has been used by Koga et a. [105, 106] to investigate the Hubbard mode with attractive interactions, but they used a sighty different scheme by directy cacuating the weights in an operator picture and appying formuas from the Hirsch-Fye [86] impurity sover in the continuum imit. This changes the weights and measurement formuas compared to the ones obtained above but eaves the samping procedure unchanged. 4.4 Spin-Mixing Formuation of CT-HYB In this section, we derive the spin-mixing version of the hybridization expansion sover, CT-HYB, which can, for instance, be used in combination with DMFT to investigate the Hubbard mode with spin-mixing hopping terms and strong interactions. This sover can be discussed very briefy after the work from the previous sections, i.e. the very detaied discussion of the CT-HYB sover in Sec. 3.3 and the derivation of the spin-mixing CT-AUX sover. The Anderson impurity mode of consideration is the same as introduced in the CT-AUX section, i.e. in equation (4.28), eading to the effective action S eff = = S 0 ( ) ( dτ(c (τ), c (τ)) τ µ 0 0 τ µ dτdτ (c (τ), c (τ)) ( Γ (τ τ ) Γ (τ τ ) Γ (τ τ ) Γ (τ τ ) dτdτ (c (τ), c (τ)) ( Γ (τ τ ) Γ (τ τ ) Γ (τ τ ) Γ (τ τ ) ) c (τ) + dτun (τ)n (τ) c (τ) ) ( ) c (τ ) c (τ ) ) ( c (τ ) c (τ ) ), (4.56) thereby defining the action S 0, which is oca in time and does not mix different spins. The corresponding Hamitonian H 0 = µ σ n σ + Un n (4.57) σ

75 4.4. Spin-Mixing Formuation of CT-HYB 71 is diagona in the oca Fock space. The CT-HYB agorithm is based on an expansion of the exponentia of the effective action (4.56) in terms of the hybridization functions. Since the Hamitonian (4.57) is diagona in the partice number of the impurity, time evoution generated by this Hamitonian wi eave the occupation of the impurity invariant. Therefore, the segment picture of the origina CT-HYB agorithm is aso a good choice for the configurations of the spin-mixing agorithm. An expansion in the hybridization wi now, due to the off-diagona terms in the hybridization, generate terms which, for a certain spin σ, contain more creation operators than annihiation operators. However, these terms wi give zero contribution to the weights, since H 0 is diagona in the partice number and the trace can be expressed in the partice number basis. Foowing the steps of the CT-HYB derivation, i.e. equations (3.79)-(3.84), it becomes immediatey cear that the weights in the segment picture can be found to be where the factor dc w C = ( 1) σ sσ e µ σ τ occ σ Uτdoube (dτ) 2(n +n ) det ( M 1 ({τ c σ, τ c σ }) ), (4.58) ( 1) σ sσ e µ σ τ occ σ Uτdoube (4.59) is the oca trace and competey equivaent to the origina CT-HYB method. In difference to the origina CT-HYB, the determinant of M 1 can not be separated into a product of determinants containing the two distinct spins ony. The meaning of the determinant is to sum up a distinct diagrams that can be drawn in the segment representation, where hybridization ines connect the starting and ending points of the segments in the impurity for given configuration, as expained in Fig The number of possibe diagrams has now been enarged from n!n! in the origina method to (n + n )! in the spin-mixing formuation of the hybridization expansion. Fig. 4.3 iustrates a set of diagrams contributing to the weight factor when spin-mixing is incuded. Figure 4.3: Iustration of 9 of the (n + n )! = 120 distinct diagrams contributing to the weight factor w C at perturbation order n = 2 and n = 3 with finite anomaous hybridization functions Γ σ σ (τ). The spin conserving (norma) hybridization functions are depicted with dotted back ines whie the spin mixing anomaous hybridization functions are drawn as dotted green ines. Fied squares indicate annihiation operators for spin-up (red), spin-down (bue), respectivey, whie empty squares indicate creation operators. The four diagrams in the vioet region are those diagrams which do not invove spin-mixing terms and therefore are the ony ones of the shown set of diagrams that woud contribute for the standard CT-HYB sover. A of the 120 diagrams for this configuration are covered by the determinant of the (n + n ) (n + n ) matrix M 1, as defined in (4.60)-(4.62). A forma derivation as in section 3.3, shows that the (n + n ) (n + n ) matrix M 1 can be expressed as M 1 ({τ c σ, τ c σ }) = M 1 ({τ c, τ c }) F ({τ c, τ c }), (4.60) F ({τ c, τ c }) M 1 ({τ c, τ c })

76 72 4. Extensions and Improvements to the CT-QMC Methods consisting of the different bocks M 1 ( M 1 σ, F, F and M 1 1. Where M is a n σ n σ matrix, defined via ({τ c σ and F σ σ is a n σ n σ matrix, with the eements ( ) F σ σ ({τ c σ, τ c σ }) σ ), τ c σ }) = Γ σσ (τ c σ i τ c σ j ), (4.61) ij ij = Γ σ σ (τ c σ i τ c σ j ). (4.62) In addition to the change in the weight factors w C, the observabes must aso be computed according to modified formuas as soon as spin mixing processes are considered. Competey anaogous to the derivation of the measurement formuas in the origina CT-HYB agorithm, we can determine the measurement formuas for the spin mixing case. By defining { 1 (τ τ β τ) = δ(τ τ τ), for τ > τ 1 β δ(β + τ (4.63) τ τ), for τ < τ as in the CT-HYB section and the inverse matrix of M 1 as consisting of the bocks ( ) M F M =, (4.64) F M where M σ is the upper-eft, ower-right n σ n σ bock and F σ σ is the upper-right, ower-eft n σ n σ bock of the matrix M, which is stored during the samping 3. The observabes for the diagona part of the interacting Green s functions is then obtained via g σσc (τ) = ij whie the off-diagona parts are determined via g σ σc (τ) = ij (τ c σ i (τ c σ i ( ) τ c σ j τ) Mσ, (4.65) ij ( ) τ c σ j τ) F σσ, (4.66) ij where the order of σ and σ is very important in the ast equaity. After determining the observabes and weights for the spin-mixing system, the samping process stays unaffected, and is described in section We can aso consider the superfuid formuation of the CT-HYB agorithm, which can be obtained, as for CT-AUX, by appying the partice-hoe transformation to the spin-down operators. This is straightforward to cacuate and is therefore not mentioned expicity within this thesis. Another important extension of CT-HYB is to consider muti-orbita impurities and aso to incude interactions not ony of density-density type but aso exchange interactions. This has been done by Werner et a. [191], for instance to appy DMFT to the Kondo attice or to mutipe band modes [114, 23]. On the other hand it is aso possibe to find a custer formuation of the CT-HYB sover, as expained by Haue et a. [82, 83] and used to investigate strongy interacting superfuid systems. However, the extensions to CT-AUX and CT- HYB introduced in this and the previous chapters have been proven to be sufficient for our investigations of topoogica phases in the modified Hofstadter mode. 4.5 Improvements for CT-HYB In this ast section of this chapter, we wi introduce some improvements for the measurement process for the hybridization expansion impurity sover, namey the orthogona poynomia representation of the observabes, as introduced in [16], and a direct measurement of the sef-energy within the CT-HYB agorithm, first discussed in [74]. These improvements wi be discussed ony for the origina CT-HYB agorithm, as derived in section 3.3, the generaization to the spin-mixing case is straight-forward. 3 One shoud note that Mσ is, of course, not simpy the inverse of M σ but instead has to be obtained by inversion of the whoe matrix M, as it are the same for F.

77 4.5. Improvements for CT-HYB Observabes in the Legendre Poynomias Basis The observabes g C (τ) in the CT-HYB agorithm have been shown to be proportiona to a δ-function in imaginary time, i.e. have to be obtained during a samping procedure according to (3.93) g σc (τ) = ij (τ c σ i τ c σ j τ)(m σ ({τ c σ, τ c σ })) ji, (4.67) with the generaized δ-function { δ(τ β (τ τ τ τ), for τ > τ τ) = δ(β + τ τ τ), for τ. (4.68) < τ To compute the observabes numericay, one has to impement a discretized imaginary time interva on a fine grid and define the δ-function on this grid. This procedure is very probematic, since the grid size has a huge infuence on the measurements. As one can easy imagine, a grid that is too coarse has a very bad resoution for the Green s functions, whereas a fine grid is very sensitive to errors coming from a finite number of Monte-Caro steps, e.g. a grid point maybe is hit too many or too few times during the simuation. Another reason that the measurement formua (4.67) is unattractive is that the statistica errors, which are unavoidabe in a MC simuation, may ead to spiky Green s functions, which are hard to hande in a combination of CT-HYB and DMFT. Therefore, it is much more convenient not to measure the Green s functions directy, but instead to measure the coefficients of the Green s functions in a poynomia representation. This was first suggested by Boehnke et a. [16] and wi be briefy reviewed in the foowing part of this section. The Legendre poynomias {L (x) N 0 } form a compete set of orthogona functions on the interva x [ 1, 1], fufiing the reations for orthogonaity and competeness 1 1 dxl (x)l m (x) = δ m (4.69) L (x)l (x ) = δ(x x ). (4.70) =0 As seen from (4.69), the Legendre poynomias are chosen to be normaized to instead of unity. This simpifies the numerica procedure, as the Legendre poynomias can be shown to fufi the recursive reation ( + 1)L +1 (x) = (2 + 1)xL (x) L 1 (x) (4.71) L 0 (x) = 1, L 1 (x) = x, (4.72) which can directy be used to compute the vaue of L (x 0 ) for x 0 [ 1, 1] for a coefficients 0 L max recursivey 4. The Green s function G σ (τ) can be expressed as G σ (τ) = g σ, L (x(τ)), (4.73) =0 with the argument x(τ) = 2τ/β 1 and the coefficients g σ, R. The coefficients are obtained by appying the orthogonaity reation (4.69) in combination with (4.73), eading to g σ, = β β 0 dτ L (x(τ))g σ (τ). (4.74) Therefore, knowing the Green s function G σ (τ) on the interva [0, β] is equivaent to knowing a the coefficients g σ, for N 0. However, in a numerica simuation, one is forced to truncate this series 4 Note that computing the poynomias on a fine grid at the beginning of a simuation and then interpoating the vaues for x 0 in every singe step is computationay not faster than to appy the recursive reations (4.71), (4.72) for every step.

78 74 4. Extensions and Improvements to the CT-QMC Methods to a finite number of these coefficients. This is not a serious probem, since it is easy to prove that the coefficients g σ, scae as ( ) 1 g σ, = O (4.75) for the case when G σ (τ) is a smooth function 5. This means that the coefficients can be restricted to 0 L max, where L max has to be chosen sufficienty arge (we use L max = 40). In a Monte-Caro simuation, the observabe of interest has been switched now from the imaginary time Green s function G σ (τ), which had to be measured on a discretized time grid without any cear estimations for the grid size, and the size of the individua grid points to the coefficients g σ,, which are discretized by definition, with the cear scaing behavior of (4.75). The Green s functions are determined by the equation C G σ (τ) = w C g σc (τ) C w, (4.76) C where C is the sum over a configurations C as expained in section Appying (4.74) to equation (4.76) directy eads to C g σ, = w C g σ,c C w, (4.77) C with the configurations coefficients g σ,c, determined via g σ,c = β = β β 0 ij dτ L (x(τ))g σc (τ) (M σ ({τ c σ β, τ c σ })) ji dτ (τ c σ i τ c σ j τ) L (x(τ)) 0 = ij (M σ ({τ c σ, τ c σ })) ji L (x(τ c σ i τ c σ j )). (4.78) Here, we defined the modified Legendre poynomias { L (x(τ)) L (x(τ)) = L (x(τ + β)) for τ > 0 for τ < 0. (4.79) The measurement of the Legendre poynomia coefficients g σ,c is numericay very simpe to impement and doesn t require much more computation time compared to the measurement of the Green s functions on a grid. However, by using a poynomia representation one aways obtains smooth Green s functions and the information that is ost by pacing a finite cut-off L max can be much better estimated than for the case of a discretized time interva in the origina formuation of CT-HYB. Except for the benchmarking cacuations of section 3.3, a cacuations using CT-HYB have been performed with measurements in the poynomia representation, where the gain in accuracy compared to the origina formuation is huge compared to effort of its impementation. For a more detaied discussion about the poynomia representation, aso invoving higher order correation functions and measurements in the Matsubara frequency representation, we refer the reader to [16] Direct Sef-Energy Measurement for the CT-HYB Agorithm In section 1.3, we saw that for the DMFT procedure both the interacting impurity Green s function G σ (iω n ) as we as the impurity sef-energy Σ σ (iω n ) are required. Whie the interacting Green s functions are obtained directy via soving the impurity probem, the sef-energy has to be determined via the Dyson equation Σ σ (iω n ) = Gσ 1 (iω n ) G 1 σ (iω n ). (4.80) 5 In [16], a scaing of O(1/) was found, which resuts from their distinct definition of the coefficients g σ,. However, for our definition of the coefficients, i.e. equation (4.74), the correct scaing is instead (4.75).

79 4.5. Improvements for CT-HYB 75 The probem with this equation in the context of a numerica Monte-Caro simuation is that G σ (iω n ) can ony be determined within statistica error. It is a we-known probem that these sma errors, when inverting the Green s function and subtracting the non-interacting Green s function, can ead to arge errors in the sef-energy and can osciate with iteration number of the DMFT process, especiay in the intermediate frequency region. To overcome this probem, one woud prefer to measure the sefenergy directy during a Monte-Caro simuation. However, this is not possibe, at east not without an unfeasiby arge effort. Fortunatey, the quantity F σ (τ) := (G σ Σ σ )(τ)/u can be measured directy without much effort, which reduces the probem of finding the sef-energy to a simpe division Σ σ (iω n ) = UF σ (iω n )/G σ (iω n ), in which the errors in both functions do not cause arger probems. In the context of the hybridization expansion sover, this procedure was first suggested by Hafermann et a. [74], however, it is a we-known feature that has been first introduced by Bua et a. [22] in the context of Numerica Renormaization Group (NRG). The derivation of the corresponding measurement formuas is rather easy, the ony requirement is the equation of motion for the interacting Green s function, which in the context of the AIM reads, dτ Gσ 1 (τ τ )G σ (τ τ ) = δ(τ τ ) + U Transforming (4.81) to Matsubara frequency representation eads to dτ δ(τ τ ) T c σ (τ )c σ(τ )n σ (τ ). (4.81) } {{ } F (τ τ ) G 1 σ (iω n )G σ (iω n ) = 1 + UF σ (iω n ), (4.82) from which we can identify UF σ (iω n ) = Σ σ (iω n )G σ (iω n ), by comparison to the Dyson equation. Therefore, in the samping procedure, we shoud aso accumuate the additiona observabe F σ (τ τ ) = T c σ (τ)c σ(τ )n σ (τ ). (4.83) In the segment picture of the CT-HYB agorithm, this observabe is amost trivia to sampe, once the interacting Green s functions are aso measured. Suppose in the Monte-Caro samping, the configuration of the present step is C. If in this configuration, the impurity is not occupied with a spin σ for the time τ, then n σ (τ ) = 0 and configuration C does not contribute to F σ (τ τ ). On the other hand, if for the configuration C the spin σ is present in the impurity, n σ (τ ) = 1 and this configuration has the same contribution to F σ (τ τ ) as it woud have to G σ (τ τ ), as one can see from (4.83). This resut aows us to determine the observabes f σc (τ), such that C F σ (τ) = w C f σc (τ) C w, (4.84) C which can just be done by impementing the indicator mentioned above ( σ is occupied, unoccupied) in the observabes for the Green s functions. The observabes f σc (τ) are then defined as f σc (τ) = ij with the indicator function (τ c σ i τ c σ j τ)(m σ ({τ c σ, τ c σ })) ji } {{ } g σc (τ) I σ (τ c σ j ) = 0 if σ is unoccupied at time τ c σ j 1 if σ is occupied at time τ c σ j I σ (τ c σ j ), (4.85). (4.86) The accumuation of the observabes f σc (τ) makes no extra computationa effort when at the same time the interacting Green s functions are measured, since the resut from this measurement is simpy mutipied by zero or unity. On the other hand, the sef-energies obtained from the division Σ σ (iω n ) = F σ (iω n )G σ (iω n ) are much more accurate and much smoother than the ones obtained via the Dyson equation (4.80) and therefore the more direct measurement of the sef-energies is much more preferabe. For further comments on improved measurement schemes for the CT-HYB agorithm, we refer the reader to [74], where improved measurements for the vertex function are aso discussed in further detai. However, for our purpose the derivation of the simpest of these improved measurement schemes is sufficient since we have not investigated more sophisticated observabes than the singe-partice correation functions.

80 76 4. Extensions and Improvements to the CT-QMC Methods

81 5. Utracod Atoms in Optica Lattices The discoveries of modern many-body cooing techniques such as aser cooing and evaporative cooing have ead to the achievement of Bose-Einstein condensation (BEC) [6, 5, 17] and the reaization of Fermi degeneracy [184, 162, 37] with utracod, diute bosonic and fermionic gases, from which a weath of research was performed. Recenty, however one of the main focuses in the fied of atomic and moecuar physics has switched from non-interacting, singe-partice physics to the exporation of correated, many-body physics, where statistics and interactions are the dominant physica effects [15]. The first step towards reaizing a quantum simuator as proposed by R. Feynman was achieved. In his semina pubication [47], Feynman discussed that a cassica computer, however impemented, wi experience an exponentia sowdown when simuating systems that are inherenty quantum mechanica in nature. He proposed to use an universa quantum simuator instead, which woud not face this probem at a. Nowadays, cod atom experiments represent a keystone for the reaization of a universa quantum simuator in two distinct ways. On one hand, they are deepy invoved in setting up a so-caed quantum computer, or quantum Turing machine, as proposed by D. Deutsch [38], which coud simuate quantum physics without the probems of exponentia sowing down by making use of the so-caed quantum paraeism. On the other hand, utracod atom experiments are very cean reaizations of mode Hamitonians, where the parameters of the underying mode can be tuned with very high precision. Therefore, simuating mode Hamitonians, such as those from condensed matter physics, with utracod atoms and exporing these quantum systems in experiment is another, more direct, reaization of a universa quantum simuator as proposed by R. Feynman. The atter idea has been reaized by oading utracod atoms into opticay created crysta attices and thereby perfecty simuating the bosonic [95, 60] and fermionic Hubbard mode [107]. Together with the tunabiity of the inter-partice interactions due to Feshbach resonances, a arge number of theoretica modes from condensed matter physics and many-body theory have been reaized, incuding for instance one-dimensiona hard-core bosons in the Tonks-Girardeau regime [103], bosons in the owest Landau eve [19] and with dipoar interactions [61], fermionic vortex attices [202] and many more. Recenty, artificia gauge fieds have been successfuy impemented in optica attice experiments [123, 124, 2, 177], simuating strong magnetic fieds with fux quanta at the order of the partice density. This has paved the way for simuating topoogica non-trivia phases, such as the quantum Ha (QH) [104, 63, 80], the quantum spin Ha (QSH) [10, 101, 110] and the fractiona quantum Ha effect [185, 116] in cod atom experiments. This deveopment is very promising towards gaining a deeper insight in the open issue of the interpay of strong correations and topoogy of the system. A great advantage of cod atom experiments with artificia gauge fieds is the perfect reaization of theoretica modes, e.g. the Hofstadter Hamitonian [87], which describes partices on a square attice subjected to a strong magnetic fied. Therefore, simuating these modes with cod atom experiments aows for a detaied comparison of theoretica predictions and experimenta observations. In this chapter, we briefy discuss the reaization of the Hubbard mode for utracod fermionic and bosonic atoms in optica attices and possibe measurement techniques, such as time-of-fight measurements and Bragg spectroscopy.

82 78 5. Utracod Atoms in Optica Lattices 5.1 Optica Lattice Potentias Neutra atoms in an osciating eectric fied wi experience an energy shift due to the AC Stark effect. This second order effect describes how the eectric fied induces an osciating dipoe moment in the atoms, which in turn interacts with the eectric fied and therefore causing the above mentioned energy shift. The induced eectric dipoe moment d(t) of an atom in an osciating eectromagnetic fied is described by d(t) = α(ω)e(t), (5.1) where α(ω) is the poarizabiity of the atom, which is a function of the frequency ω of the eectric fied E(t). The resuting conservative potentia is determined via [62] V dip = 1 2 Re d E t = 1 2 Re(α(ω)) E2 t, (5.2) where the time average... t eads to a static dipoe potentia in the imit of stationary atoms (which is reasonabe when the time-scae for the center-of-mass motion of the atom is much sower than the osciating frequency of the eectric fied). The poarizabiity of the atom is commony discussed in an (effective) two eve system, subjected to a cassica eectromagnetic fied. In this mode, the excited state has an energy difference ω 0 with respect to the ground state and a finite decay rate Γ e, which eads to the poarizabiity being a compex number. Due to this, the conservative part of the potentia is determined by the rea part of α, as in (5.2). In terms of the effective two eve system, the potentia (5.2) can be expressed as [62] V dip = Ω Rδ δ 2 + Γ 2 I(x). (5.3) e/4 Here, Ω R is the Rabi frequency associated with the effective mode, δ = ω ω 0 is the detuning of the eectric fied with respect to the energy difference of the two eve system and I(x) is the time averaged intensity of the eectric fied. Depending on the sign of δ, the potentia can be chosen to be attractive or repusive by varying the frequency of the eectric fied 1. Commony, the different detunings are denoted as red for δ < 0, i.e. for V dip < 0 being attractive, and bue for δ > 0, i.e. for V dip < 0 being a repusive potentia. To create an optica attice potentia in a certain direction (which we denote as the x-direction), two counterpropagating aser beams with identica frequency and intensity are spatiay overayed. For equa poarization vectors, this resuts in E 2 (r, t) = 4E 0 cos 2 (ωt) cos 2 (k x x), (5.4) with the eectric fied strength E 0 and wave vector k x. Time-averaging of the cos(ωt) eads to a factor 1/2, such that the attice potentia in x-direction is described by V dip (r) = s E R cos 2 (k x x) = s E R cos 2 (πx/a x ), (5.5) where we introduced the dimensioness attice strength s, the recoi energy 2 E R = 2 π 2 /2ma 2 x and the attice constant in x-direction a x, which determines the ength of the unit ce of the attice. To construct a two- or three-dimensiona attice potentia, additiona pairs of counterpropagating aser beams can be added in the y- and or z-direction, eading to an optica attice potentia V dip (r) = s x cos 2 (πx/a x ) + s y cos 2 (πy/a y ) + s z cos 2 (πz/a z ), (5.6) where we aow for different strengths s i and attice constants a i for the different directions. In the previous discussion of the optica attice potentia, the aser beams which create the attice potentia in the x-direction have been considered to be infinitey extended in the y- and z-direction. However, these aser beams typicay have a Gaussian shaped intensity profie in the pane orthogona to the propagation direction. Since the potentia (5.2) is proportiona to the aser intensity, it has aso a Gaussian shape. 1 However, the detuning aso infuences the imaginary part of α, which induces a dissipative part of the potentia and scaes as δ 2. To maximize the ife-time of the physica systems in experiments, the detuning has to be chosen such that the dissipative effects can be negected for sufficienty ong time-scaes. 2 In the foowing, E R wi be set to unity.

83 5.2. Interactions and Feshbach Resonances 79 Therefore, the systems that can be simuated using utracod atoms in combination with optica attices are finite systems, that are too sma to be considered as being in the thermodynamic imit. Depending on the sign of the attice potentia, the finite aser beams ead to an additiona confining (red detuned asers) potentia or to a repusion of the atoms from the center (bue detuned asers) 3. In any case, the simuated systems are confined to a finite region in space, typicay modeed with an additiona harmonic potentia, eading to 3 3 V dip (r) = s i cos 2 (πx i /a xi ) + V i x 2 i = V att (r) + V trap (r), (5.7) i=1 where V att is the bare attice potentia and V trap the trapping potentia, which is the eading order term of the Gaussian potentia 4. For many physica effects, this trapping potentia must be incuded for both theoretica and experimenta investigations and we wi discuss its effect on topoogica insuating systems in Sec For the content of this thesis, it is aso reevant that in optica attice experiments attice modes with dimensions ower than three can be simuated, as it is for instance necessary to investigate the integer quantum Ha effect. This is done by appying a two-dimensiona attice potentia and confining the system in the residua direction, for instance by a dipoe trap. If the energy gap between the excited states and the ground state in the trap is much arger than the temperature of the system, it is effectivey confined to two dimensions, since excitations in the third direction are very unikey to occur. Another possibiity to create two-dimensiona systems is to use a three-dimensiona attice as described by (5.5) but to increase the strengths of the attice potentia in one direction such that tunneing in this direction is severey reduced. The resuting setup is then an array of two-dimensiona ayers, where each ayer represents a two-dimensiona optica attice. Both methods have been successfuy used to experimentay expore the physics in two-dimensiona attice systems [14]. i=1 5.2 Interactions and Feshbach Resonances The interest in optica attice experiments with utracod atoms resuts not ony from the high tunabiity of the attice potentia to which the partices are exposed to but aso from the abiity to precisey tune the inter-partice interactions, aowing for the experimenta simuation of many-body effects, such as coective excitations or spontaneous ordering in the many-partice system. The (two-partice) interactions between neutra atoms resut from the short-ranged van der Waas force, which shows a characteristic 1/r 6 decay for sufficienty arge distances r between two partices. Commony, in the imit of diute gases and very ow temperatures, the interactions between partices are described within the framework of scattering theory. For sufficienty ow energies and densities, an expansion of the two-partice scattering process in terms of partia waves reveas that when the energy scaes invoved in the scattering process are sufficienty ow, ony s-wave scattering dominates. In this case a higher order contributions are negigibe and the interaction is sufficienty we described by the s-wave scattering ength a s [55, 15]. The s-wave state is spatiay symmetric, which impies that two bosonic partices of the same interna quantum state can interact with each other, whie for fermions ony two partices in distinct interna quantum states can interact because of the symmetric, anti-symmetric two-partice wave functions, respectivey. The s-wave scattering ength can theoreticay be obtained by taking the imit a 2 σ tot = im k 0 4π, (5.8) where σ tot is the tota eastic cross section and k is the difference in momentum of the two partices [113]. However, since the exact interaction potentia between two atoms is ikey unknown and very hard to determine, it is much more convenient to measure the scattering ength directy in experiments. The interaction potentia can be modeed by an arbitrary pseudo potentia, which is required to reproduce the correct scattering ength in the ow energy imit. Commony, the pseudo potentia is chosen as [55] V (r r ) = 2πa s m δ(r r ), (5.9) 3 In this case, additiona trapping potentias are needed, as for instance optica dipoe traps. 4 By overaying different aser beams, more compicated trapping potentias are possibe, however, the harmonic potentia is the common approximation for deep attices.

84 80 5. Utracod Atoms in Optica Lattices with the reduced mass of the partices m. An important too to experimentay tune the strength of the inter-partice interactions independenty of other experimenta parameters is the use of Feshbach resonances. By making use of Feshbach resonances, it is possibe to adjust the scattering ength to very arge vaues compared to the inter-atomic distance or to change the sign of a s, thereby making the partices attracting each other, by appying an additiona magnetic fied. The occurrence of Feshbach resonances is due to the fact that the scattering partices are atoms and therefore have many interna degrees of freedom aowing for ineastic scattering processes. Figure 5.1: Scattering ength between two hyperfine states of 6 Li as a function of the externa magnetic fied B. The Feshbach resonance is ocated at B 0 = 834G and the width of the resonance is approximatey B = 300G. The off-resonant scattering ength a s0 = 1405a B (a B is the Bohr radius) is exceptionay arge for this configuration. Figure from Ref. [15]. In a two-atom scattering process, whenever the energy associated with an eastic scattering process (referred to as open channe) approaches the energy of a bound state (referred to as cosed channe), the two atoms can temporariy form a bound state, which significanty infuences the scattering properties. For the theoretica treatment of these effect, a singe-channe anaysis, i.e. considering eastic scattering processes ony, is not sufficient and one has to consider a muti-channe scattering probem [176]. For the muti-channe probem, the resuting scattering ength depends on the energies of the different channes [181, 183, 137], which in turn can be tuned by an externa magnetic fied B. The resuting scattering ength can approximatey be described by the formua ( a s (B) = a s0 1 B ), (5.10) B B 0 where the effective parameters B and B 0 are being considered as the width of the resonance and the fied strength at resonance, respectivey [137], and have to be determined in experiment, as we as the off-resonant scattering ength a s0. In Fig. 5.1 magnetic fied dependence of the scattering ength is shown for the case of 6 Li. 5.3 Hubbard Parameters for Optica Lattices We have discussed the singe-partice potentia set up by the asers, i.e. described by equation (5.7), and the effective interaction potentia (5.9). However, many-body physics is commony expressed in the second quantization framework, therefore we now briefy discuss the second quantized Hamitonian of the interacting system, which is the Hubbard Hamitonian, discussed in section 1.1, and aso the Bose- Hubbard Hamitonian, describing interacting fermions or bosons, respectivey. Equation (5.7) expresses

85 5.3. Hubbard Parameters for Optica Lattices 81 the singe-partice potentia of the system in terms of a trapping potentia V trap potentia V at, the atter being defined as and a periodic attice V at (r) = 3 s i cos 2 (πx i /a xi ), (5.11) i=1 in units of the recoi energy E R. In the regime of sufficienty deep attices, i.e. s 5 the attice potentia is the dominant potentia for determining the physics in the center of the trap, i.e. for sufficienty sma partice numbers in the system. Because of this, the Hamitonian of the system is convenienty expressed either in terms of Boch states α, k, which are the eigenstates of the Hamitonian H at = p2 2m + V at (5.12) and abeed with the band index α N and the quasi-momentum k i [ π/a i, π/a i ], or in terms of Wannier states α,, which are obtained from the Boch states via Fourier transformation α, = 1 e ikr αk (5.13) L and for appropriate choice of phase factors are maximay ocaized around attice site 5. The eigenvaue equation for the attice Hamitonian then reads k H at α, k = ɛ α (k) α, k, (5.14) where ɛ α (k) is a smooth function in terms of the quasi-momentum k, but is we separated in energy for different band indices, i.e. features band gaps α,α min ɛ α (k) ɛ α (k) > 0 for α α. This makes the set of Wannier states a good basis, since athough they mix a possibe quasimomenta, they respect the separation of the system into distinct bands. Now, incuding the spin σ as a abe for the partices interna degree of freedom, the singe-partice Hamitonian H = p2 2m + V at + V trap (5.15) can be expressed in the basis of Wannier states α,, σ according to H = α,, σ t α,m α, m, σ + α,, σ V α,α,m α, m, σ, (5.16) α,,m,σ α,α,,m,σ with the matrix eements of the attice Hamitonian ( ) p t α 2,m = α,, σ 2m + V at α, m, σ = k e ik(r R m) ɛ α (k), (5.17) which respect the separation of the system into distinct bands 6, and the matrix eements of the trapping potentia V α,α,m = α,, σ V trap α, m, σ, (5.18) which are not diagona with respect to the band index α. Here, the aser potentias that we are considering do not make a distinction for the different interna states of the partices and therefore there is no additiona dependence on σ in the matrix eements of the Hamitonian 7. When the band gap 1,α 5 Maximay ocaized in this context means that the Wannier states α, are the most ocaized states in the subspace of fixed band index α. 6 The minus sign in (5.17) is convention, eading to the matrix eements being positive. 7 There exist experiments which make use of optica potentias, which distinguish between the different hyperfine states of the atoms and therefore woud have additiona dependence on σ in the matrix eements of the Hamitonian. A corresponding mode is discussed in Sec. 5.4.

86 82 5. Utracod Atoms in Optica Lattices between the first band (α = 1) and a higher bands α > 1 is much arger than the typica interaction energy and the temperature of the system and the trapping potentia is very fat 8, it is sufficient to express the resuting ow energy physics in a owest band approximation, i.e. negecting a contributions from α > 1. This approximation reies on the fact that the energeticay we separated quantum states in the higher bands are ony sparsey popuated, thereby having no infuence on the physics of the system, which is dominated by the hugey popuated states in the first band 9, therefore we ony consider quantum states from the owest band and drop the index α further on. The Wannier states, σ are ocaized around attice site and decay exponentiay with the distance, which eads to the matrix eements t,m and V,m in equation (5.16) aso decaying exponentiay in the distance m. For the case of optica attice potentias this decay is so strong, that taking ony nearest neighbor matrix eements, i.e. those with m = 1 into account, is a perfecty reasonabe approximation [95]. For the matrix eements of the trapping potentia (5.18), it is sufficient to ony consider the oca matrix eements, where = m, since the non-oca matrix eements decay even faster than for the hopping matrix eements, caused by the ack of a "non-oca" operator ike the momentum operator in (5.17). Combining a the discussed simpifications and approximations and subsequenty switching to the anguage of second quantization, the Hamitonian of the system without interactions reads H = t,m c σ c mσ + V, c σ c σ, (5.19),σ,m,σ where the creation and annihiation operators c σ, c σ create or annihiate a partice with spin σ at attice site and, m means that ony the sum over nearest neighbors, m is evauated. The non-interacting Hamitonian (5.19) is the same for both bosonic and fermionic partices in an optica attice. The nature of the partices ony enters the Hamitonian in form of the different commutation reations of the creation and annihiation operators, i.e. the fermionic operators anti-commute whereas the bosonic operators commute {c σ, c mσ } = c σ c mσ + c mσ c σ = δ mδ σσ, (5.20) [c σ, c mσ ] = c σ c mσ c mσ c σ = δ mδ σσ. (5.21) As we saw in the previous section, bosonic atoms with the same interna quantum state can interact via s-wave scattering, whie for fermions ony atoms with different interna quantum states can interact with each other via s-wave scattering. Therefore the simpest interacting bosonic systems are those having no interna degrees of freedom, whie for fermions the simpest systems are those which are described by atom with two possibe interna quantum states, which can be seen as the spin of the partice σ =. For the bosonic system (without interna degrees of freedom), the interacting part of the Hamitonian is described as H Int = 1 U mst c 2 c mc t c s, (5.22) whereas for fermions, it reads,m,s,t H Int = 1 U mst c σ c 2 mσc tσ c sσ, (5.23),m,s,t,σσ with the matrix eements U mst = 2πa s m w (r)w m(r)w s (r)w t (r)dr, (5.24) according to (5.9) the same for bosonic and fermionic partices. For sufficienty deep attices, the Wannier functions w (r) = r at site again have amost no overap with their neighboring sites, such that it 8 A of which is fufied in optica attice experiments [15]. 9 For a discussion of the infuence of second order processes on the matrix eements in a owest band approximation see for instance [13, 127]

87 5.4. Artificia Gauge Fieds for Neutra Atoms in Optica Lattices 83 is reasonabe to ony take into account the purey oca parts of the interaction, i.e. the matrix eements with = m = s = t. Then the interacting Hamitonian reads H Int = U n (n 1) (5.25) 2 for bosonic partices and H Int = U n n, (5.26) for fermionic partices. The Hubbard interaction parameter U is then defined according to U = 2πa s w (r) 4 dr. (5.27) m Adding the interactions to the singe-partice Hamitonian (5.19), we obtain the inhomogeneous Bose- Hubbard mode H = t,m c c m + V, c c + U n (n 1) (5.28) 2,m for a system of interacting singe-component Bosons and the inhomogeneous Fermi-Hubbard mode H = t,m c σ c mσ + V, c σ c σ + U n n (5.29) σ,m,σ for a system of interacting two-component Fermions in optica attices. Indeed, despite the approximations made to obtain (5.28) and (5.29), optica attices fied with one species of interacting bosons have been shown to be a near perfect reaization of the Bose-Hubbard mode [95, 60], i.e. of (5.28). Simiary the same statement hods for a two-species system of interacting fermions and (5.29), as shown in theory and experiment, for instance in [90, 159, 88]. 5.4 Artificia Gauge Fieds for Neutra Atoms in Optica Lattices Utracod Atoms in Optica Lattices can be seen as perfect experimenta reaizations of theoretica modes from condensed matter physics, such as the fermionic or bosonic Hubbard mode, as we have seen in the ast chapter. This was proven by many experiments, which resoved for instance the Fermi-surface of non-interacting Fermions [45, 107], the metaic and Mott-insuating phase of interacting Fermions [98, 159] and the superfuid-mott insuator transition for interacting Bosons [60]. An important cass of theoretica modes are those, which expain the emergence of topoogicay non-trivia phases, such as the quantum Ha effect or the quantum spin Ha effect, which we wi discuss in the foowing chapter. Such modes are for instance the Hofstadter mode [87], the Kane-Mee mode [99] or the Bernevig- Hughes-Zhang mode [9]. A of them have in common, that the hopping parameters corresponding to the kinetic energy in these modes show compex phases, corresponding to a non-trivia gauge potentia. The gauge potentia resuts either from externa magnetic fieds, which eads to a spatiay dependent but Abeian gauge potentia or from intrinsic spin-orbit couping, for instance of Rashba type, which eads to homogeneous but non-abeian gauge potentias. The experimenta simuation of these theoretica modes with cod atoms is rather compicated, as, on one hand, the atoms themseves are neutray charged such that an externa magnetic fied does not affect the hopping of the partices at a and, on the other hand, the atoms do not show any kind of natura spin-orbit couping, such that for both effects, the gauge potentia has to be introduced to the atoms artificiay [96, 143, 54, 34]. This eads to the topic of artificia gauge fieds for neutra atoms, which has gained much interest because of the promising possibiity to investigate topoogica phases in cod atom experiments where, in contrast to condensed matter, a experimenta reevant parameters can be controed from the experimentaist. Since the reaization of artificia gauge fieds has come into focus of theorists and experimentaists, a pethora of pubications has proposed how to impement severa kinds of

88 84 5. Utracod Atoms in Optica Lattices gauge potentias in cod atom experiments [123, 124, 119, 28, 57, 96, 143, 54, 34], reaching from the creation of artificia effective magnetic fieds, to optica fux attices and even time-reversa invariant non-abeian gauge fieds. Here, we wi focus on a proposa by Godman et a. [57], which focuses on the reaization of a combination of Abeian and non-abeian artificia gauge fied for a two-component fermionic system on an atom chip. The mode proposed in this etter is the mode that the theoretica investigations in the foowing chapters wi be focused on. The proposed setup uses a combination of Raman-assisted tunneing and an aternating Zeeman attice, which we wi expain in the foowing Raman Transitions in the Λ-system Consider an atomic system consisting of three interna states, two ow energy states { g 1, g 2 } with energies ω 1, ω 2 ying cose to each other and a high energy excited state e with a much arger energy ω e, where the two ow energy states are couped to the excited state via two far off-resonant aser beams. The frequency of the first aser is denoted as ω1 L = ω e ω 1 + δ 1, where δ 1 is the detuning from the transition frequency, whie the frequency of the second aser can be written as ω2 L = ω e ω 2 + δ 2, with the second detuning δ 2. It is more convenient to switch to a rotating frame [25], which simpifies the view on the situation Figure 5.2: Iustration of the Λ-system in an energy eve diagram. The two ow energy states g 1, g 2 are separated by an energy difference δ and the excited state e is separated from both states by the energies ± δ/2. introduced above. In a rotating frame, the two ow energy states are separated by an energy difference δ = δ 2 δ 1 and the excited state is separated from the states g 1, g 2 by the energies ± δ/2, where = δ 2 + δ 1, see Fig After performing a rotating wave approximation (see for instance [151]), an effective, Rabi-type mode is obtained, which is described by the Hamitonian H Rabi = 1 2 δ 0 Ω a 0 δ Ω b Ω a Ω b 2, (5.30) where Ω i, i = a, b is the corresponding Rabi-frequency Ω i = Ω i e iφi with the spatiay dependent phase φ i = k i r. In the foowing, we consider >> δ, Ω i, i.e. very arge detuning of the asers. Then, starting with a state ψ without any popuation of the excited state, the excited state wi stay amost competey unpopuated during the time evoution. Expoiting this fact, an effective mode for the ow energy states ony can be derived. Assuming a state ψ(t) = α(t) β(t) γ(t), (5.31)

89 5.4. Artificia Gauge Fieds for Neutra Atoms in Optica Lattices 85 the Schrödinger time-evoution of this state is determined according to i t α(t) β(t) = H Rabi α(t) β(t) = 1 δα(t) + Ω aγ(t) δβ(t) + Ω b 2 γ(t) γ(t) γ(t) 2 γ(t) + Ω a α(t) + Ω b β(t) For stationary γ(t), i.e. t γ = 0, the third row of (5.32) requires. (5.32) γ(t) = Ω a 2 α(t) Ω b β(t), (5.33) 2 which aows us to repace γ(t) by α and β 10. Repacing γ by (5.33) in equation (5.32) eads to an effective time-evoution for the α and β coefficients. ( ) ( ) α(t) i t = β(t) H α(t) Rabi, (5.34) β(t) with the simpified Rabi Hamitonian H Rabi = 1 2 where the Rabi frequencies Ω R are defined as ( δ + Ωa 2 2 Ω R Ω R δ + Ω b 2 2 ), (5.35) Ω R = Ω aω b 2 = Ω R e i(ka k b)r, (5.36) eading to the Hamitonian in (5.35) being spatiay dependent. Typicay, the aser beams which drive the transitions from the ow energy states to the excited states have to be chosen to account for the interna structure of the atoms, e.g. for different anguar momenta of different intera states, which was not discussed here. A set of aser beams reaizing the Hamitonian (5.35) for a set of two distinct interna states g 1, g 2 of an atom is caed a Raman setup, or a set of Raman asers. The transitions are caed Raman transitions Zeeman Lattice pus Raman Beams The system we consider consists of atoms with two different hyperfine states g 1, g 2 11, which are confined to two dimensions (see section 5.1). In the x-direction, two aser beams are creating a attice with attice potentia V x (x) = s x cos 2 (πx/a x ), (5.37) which acts equay on both hyperfine states. In the y-direction, a Zeeman attice is buid by impementing stripes of constant magnetic fied aong the x-direction but with aternating sign in the y-direction [57]. The both hyperfine states, due to their distinct interna spin, fee a attice potentia in the y-direction. In contrast to the attice in x-direction, the attice in the y-direction is state dependent, i.e. g 1 fees a maximum in the potentia at the pace where g 2 fees a minimum and vice versa. Additionay, there is a tota energy shift B between the two hyperfine states. In the proposa of [57], the Zeeman attice is created on an atom chip by current carrying wires, oriented in the x-direction, such that the current traves in the opposite direction in two neighboring wires. Approximating the Zeeman attice by a cosfunction, which is sufficient for the tight-binding approximation, then eads to the attice potentia V (x, y) = V x (x)1 + V y (y)σ z + Bσ z, (5.38) 10 It is evident that the choice of γ in (5.33) is itsef time dependent and therefore not sef-consistent with the assumption [25, 94]. However, the approximation is we estabished since it eads to resuts in good agreement with exact soutions. For a detaied anaysis, see [20] 11 There must be more than two interna states for these atoms, however, we consider a others to be essentiay unpopuated.

90 86 5. Utracod Atoms in Optica Lattices where σ z is the Paui z-matrix, acting on the subspace of the two hyperfine states and V y (y) = s y cos 2 (πy/a y ) (5.39) is the potentia for the first hyperfine state. To create a square attice in the end, we chose a y = 2a x = 2a. The Hamitonian for the Zeeman attice configuration then reads H 0 = p2 2m 1 + V x(x)1 + V y (y)σ z + Bσ z. (5.40) In the x-direction, a partice can hop from one attice site to the next attice site without changing the interna state and therefore the system is described by a tight-binding mode in the x-direction. On the other hand, hopping from one attice site to a neighboring site in the y-direction requires the change of the partice s hyperfine state because of the aternating potentia in this direction. Without changing the hyperfine state, the partice can ony hop to the next nearest neighbor, which is strongy suppressed due to a sma spatia overap and therefore, without additiona potentias that change the hyperfine states of the partices, no hopping in y-direction takes pace. To aow for tunneing in the y-direction, additiona operators are required, which change the interna state of the partices. Equation (5.35) exacty describes such an operator, which is the operator describing two Raman beams acting on the system. Adding these two Raman beams, eads to the tota Hamitonian of the system H = p2 2m 1 + V x(x)1 + V y (y)σ z + Bσ z + H Rabi (5.41) which contains diagona and off-diagona terms in the hyperfine states. The corresponding tight-binding Hamitonian in second quantization can now be expressed as H = i,j,σ,σ t σσ ij c iσ c jσ, (5.42) where i, j abe the attice sites of the two-dimensiona system and the sum is running ony over pairs of nearest neighbors. The index σ = g 1, g 2 abes the two hyperfine states. For hopping in the x-direction, ony the hopping ampitudes between equa hyperfine states are non-zeros, whie for the y-direction ony hopping ampitudes between distinct hyperfine states are non-zero, due to the Raman couping. As shown in (5.36), the off-diagona terms in the Hamitonian (5.42) carry a position dependent phase factor e i(ka kb) r, which can be adjusted by changing the wave-vectors of the Raman beams k a,b. To reaize phase factors which break time-reversa symmetry, i.e. reaize quantum Ha physics, one can reaize the Landau gauge by adjusting the phase factor such that they depend on the x-coordinate ony. Then the hopping matrix eements in the x-direction woud obtained by the common formua t x ij = w(r)h 11 w(r + a x e x )dr = t x, (5.43) with H 11 being the first matrix eement of the Hamitonian (5.41) and e x the unit vector in x-direction. Therefore, the hopping eements in the x-direction can be chosen purey rea and are uniform. On the other hand, the hopping matrix eements in the y-direction are defined according to t y ij = w(r)h 12 w(r + a y /2e y )dr = Ω R 2 w(r)e i(ka k b)x w(r + a y /2e y ), (5.44) where we have aready expoited that the Raman beams are chosen such that the phase factor ony depends on the x-coordinate. Soving the equations above eads to the hopping matrix eements for the y-direction being t y ij = ty e i2παxi (5.45) with uniform magnitude and spatia dependent phase e i2παxi, where α = (k a k b )a y /2π = (k a k b )a/4π. The absoute vaue of the hopping ampitude in x- and y- direction can be tuned independenty, aowing the reaization of the desired case of t y = t x t by adjusting the experimenta parameters. On the other, the fux parameter α can be adjusted the same way, aowing for the specia cases, where

91 5.5. Time-Reversa Invariant Topoogica Insuators with Cod Atoms 87 α = p/q with p, q N and therefore reaizing the ceebrated Hofstadter mode with cod atoms 12. The two-component system introduced here can effectivey be seen as a singe-component system, since, due to the fact that the two hyperfine states fee an opposite potentia in the y-direction, a given coumn in the x-direction wi ony be popuated by a singe hyperfine state. In other words, every even coumn wi be occupied with atoms in the hyperfine state g 1, whie every odd coumn wi be popuated ony with atoms in the hyperfine state g 2, such that the doube occupancy of every attice site is zero, due to the Paui principe. This eads to the fact that the system can be treated as an effective singe partice system of atoms in just a singe interna state, which we ca g. 5.5 Time-Reversa Invariant Topoogica Insuators with Cod Atoms In the previous section, we introduced a possibe reaization of artificia Abeian gauge fieds for utracod atoms in optica attices. In this section, we discuss very briefy how this system can be generaized to two-component systems, incuding non-abeian gauge fieds. The Hamitonian, which is to be reaized, is the time-reversa invariant generaization of the Hofstadter mode with two component fermions, in second quantization stating H = t x,y c x+1,y ei2πγσx c x,y + c x,y+1 ei2πασzx c x,y + h.c.. (5.46) Here, we introduced the vector notation of the creation operators on a attice site at coordinates (x, y), c x,y = (c x,y,, c x,y, ) and the rea parameter γ which tunes the non-abeian phase, proportiona to the Paui matrix σ x, which mixes the spin. The spatia dependent phase e i2πασzx has opposite sign for both components, which eaves the Hamitonian time-reversa symmetric and therefore aows for quantum spin Ha physics. After introducing the Hofstadter mode in the ast section, the experimenta reaization of its time-reversa invariant generaization (5.46) is, in principe, straight forward. Figure 5.3: Iustration of the Raman assisted tunneing in the y-direction of the attice. The Raman beams coupe either the states g 1 and g 2 or the states e 1 and e 2. The attices for both components are aternating, such that tunneing ony happens due to the Raman beams, which imprint a phase to the partice according to (5.36). Figure from Ref. [57]. The first step is to experimentay reaize a copy of the Hofstadter system with a set of new hyperfine states e 1, e 2, couped by two distinct Raman beams in the same attice as the origina system. Then the 12 For a discussion of the Hofstadter mode and how the spatiay dependent phases effect the singe-partice physics of the system, we refer to the next chapter. The aim of this chapter was simpy to expain one experimenta reaization of artificia gauge fieds in optica attice experiments on a very basic eve.

92 88 5. Utracod Atoms in Optica Lattices fuxes α, α of the two systems can be tuned independenty, and one can reaize the specia case α = α. If these two systems are uncouped but ocated in the same attice, the Hamitonian (5.46) is reaized for the specia choice of γ = 0. To tune the parameter γ to vaues arger than zero, it is sufficient to aso aow for Raman assisted tunneing in the x-direction by choosing an appropriate phase of the Rabi frequencies (5.36) for the corresponding transitions. An iustration of the system with γ = 0 is depicted in Fig. 5.3, taken from Ref. [57], aso showing reaistic experimenta parameters. The experimenta schemes to achieve artificia gauge fieds in optica attices presented in this chapter have been introduced on a very basic eve, without going into any detais of the current experimenta issues, that emerge when trying to reaize these gauge fieds within optica attices. For more detais about the experimenta reaization of certain casses of gauge fieds and the path towards the reaization of more genera systems, we refer the reader to [123, 124, 125, 120, 2, 177].

93 6. Z-Topoogica Insuators in Optica Lattices Z-topoogica insuators are known as buk insuating systems, which can be cassified in terms of a Z quantum number ν. For ν = 0 these systems are topoogicay equivaent to an ordinary insuator and therefore topoogicay trivia. On the other hand for ν 0 these systems are topoogicay equivaent to a quantum Ha system with the corresponding topoogica quantum number. A reaizations of Z topoogica insuators can be discussed in terms of the corresponding quantum Ha system. In this chapter, we wi introduce the quantum Ha effect on a very genera eve, without focusing on a specific Hamitonian, and discuss the Chern topoogica invariant and the buk-boundary correspondence. After that, we introduce the Hofstadter mode, which is a attice mode in tight-binding approximation that is abe to show quantum Ha phases. The main part of this chapter wi then be the discussion of the reaization of Z topoogica phases with utracod atoms, which is pubished as an artice [21]. 6.1 The Quantum Ha Effect In the Quantum Ha effect, the transverse Ha conductivity of a two-dimensiona eectronic system subjected to high magnetic fieds is found to be quantized according to σ xy = Ne 2 /h, where N is a topoogica invariant integer, which is caed the tota Chern number of the occupied bands [182, 104, 141]. Since its first experimenta detection [104], the quantum Ha effect has been estabished as the textbook exampe of a topoogica non-trivia insuator, i.e. an insuator that is, from a topoogica point of view, not equivaent to the vacuum state which woud be the case for an ordinary band insuator [78]. In this section, we wi discuss the quantum Ha effect from a topoogica point of view by briefy reviewing the TKNN invariant [182] and Berry phase [11], and aso discuss the emergence of gapess boundary states, so-caed edge states, at interfaces where the topoogy of the system changes TKNN Invariant and Berry Phase for the QHE The difference between an ordinary insuator (ordinary refers to the conductivity tensor being identica to zero) and a quantum Ha state in terms of topoogy has been expained by Thouess et a. [182] (TKNN) in a semina paper from 1982, which we wi shorty review here. The essentia resut of the TKNN paper is that the transverse Ha conductance σ xy for a buk insuating system can be expressed as σ xy = N e2 h, (6.1)

94 90 6. Z-Topoogica Insuators in Optica Lattices where N N is an integer, the so-caed tota Chern number of the occupied bands, which is a topoogica invariant. In inear response theory, the transverse Ha conductivity σ xy of a fermionic attice system is described by the Kubo formua 1 σ xy = i V n>0 ψ n j x ψ 0 ψ 0 j y ψ n (E n E 0 ) 2 ψ 0 j x ψ n ψ n j y ψ 0 (E n E 0 ) 2, (6.2) where ψ n, ψ 0 abe the many-partice excited states and ground state, E n, E 0 are the corresponding eigenenergies of the attice Hamitonian and j α, α = x, y is the current operator for the α-direction, see for instance [32]. TKNN expressed the Ha conductivity in terms of the eigenstates of the system, which are the Boch states u α (k) with band index α and quasi-momentum k B, where B is the first Briouin zone. The expression they found for the Ha conductivity (in a more convenient formuation) states σ xy = e2 2πh ɛ α<ɛ F B dki ( kx u α (k) ky u α (k) ky u α (k) kx u α (k) ), (6.3) where the integra runs over the compete Briouin zone and the sum covers the occupied bands. The quantization of the Ha conductivity then comes from the fact that the integrand is the so-caed Berry curvature, which is an anti-symmetric second rank tensor or, in other words, a 2n-form (with n = 1) over a two-dimensiona space M (which we eave unspecified). According to the Chern-Gauss-Bonnet theorem [31], the integra over the compete Briouin zone in (6.3) returns the vaue σ xy = e2 2πh ɛ α<ɛ F 2πχ(M α ) (6.4) where χ(m α ) is the Euer characteristic of the subset M α, which is the αth magnetic Briouin zone of the system, embedded in the first tota Briouin zone. Since the Euer characteristic is aways an integer, i.e. χ(m α ) = n α N 0, the Ha conductivity finay reads σ xy = e2 h N, with N = α n α. (6.5) The TKNN paper has two essentia resuts, first the Ha conductivity is proven to be stricty quantized as soon as the Fermi energy of the system is ocated in a buk gap and as ong as inear response theory is appicabe and, second, the mathematica reason for the quantization of the Ha conductance is deepy reated to the topoogy of the of the Boch functions u α (k) and therefore of the Hamitonian that describes the physica system. The topoogica properties of an underying system have to be understood as goba properties, which can not be easiy changed by sma perturbations of the system (in contrast to geometrica properties, which are defined ocay) and therefore the quantization of the Ha conductance is very robust against externa perturbations which cannot change the topoogy [182]. The Berry curvature F α (k) = i ( kx u α (k) ky u α (k) ky u α (k) kx u α (k) ) (6.6) in Eq. (6.4) is cosey reated to another very important physica quantity, namey the Berry phase (or geometric phase) γ α [11], which is defined as the cosed path integra over the first Briouin zone γ α = A α (k)dk = i u α (k) k u α (k) dk. (6.7) As one directy sees, and therefore, by appying Stoke s theorem γ α = A α (k)dk = A α (k) = i u α (k) k u α (k) dk = F α (k) (6.8) B A α (k)dk = B F α (k)dk. (6.9) 1 Here we restrict ourseves to the rea part of the zero frequency conductivity, i.e. σ xy Re(σ xy(ω = 0)).

95 6.1. The Quantum Ha Effect 91 This, according to (6.3), reates the Berry phase γ α directy to the Ha conductance σ xy. The Berry phase aows for a physicay more appeaing expanation of the quantized Ha conductance, as we wi sketch shorty. Suppose there are no degeneracies aong a cosed oop in the Briouin zone, such that the Boch functions u α (k) accumuate a we defined phase e iφ when the quasi-momentum k is transported adiabaticay aong this cosed oop in the Briouin zone. The accumuated phase is then identica to the Berry phase, i.e. φ = γ α, and must be an integer mutipe of 2π such that the states are eft unchanged after the adiabatic evoution aong a cosed path [182]. In the TKNN paper, the quantization of the Ha conductance was expained by using Berry s phase rather than the Chern-Gauss-Bonnet theorem. Whie using the atter eads to a more rigorous proof [141] of the quantization, it aso requires much more mathematica background on the theory of fiber bundes on compex vaued manifods. The TKNN invariant (6.5) nicey describes the quantization of the Ha conductance σ xy in cases when the Fermi energy is ocated in a buk gap of the infinite system. However, it requires a we defined Briouin zone, which ony exists for transationay invariant, infinitey sized systems. For the theoretica description of finite system the so-caed edge states pay an essentia roe, which are states that are energeticay ocated in the buk gap of the infinite system and spatiay ocaized aong the boundary of the finite system, thereby carrying the Ha current aong this boundary. In the next section we wi focus on finite systems and discuss the emergence of these edge states and their reevance for the QHE and topoogica insuators in genera Topoogica Edge States and the Buk Boundary Correspondence In the previous section, we identified the quantum Ha effect as a topoogica feature of an infinite system. To drive a quantum Ha system, or more genera, a topoogica non-trivia system, into a topoogicay trivia system, the topoogy of the system must be changed. In genera, changing the topoogy is possibe by aowing buk bands, which were formery we separated by a buk gap such that the Chern number of both bands was we defined, touch each other at certain points in the Briouin zone. When two bands touch each other, the Chern number, i.e. the topoogy, of the individua bands is no onger we defined. Further deformation of the system wi ead to spitting of the bands, such that the Chern number is we defined again but may have changed during the process of deformation. Now imagine a finite topoogicay non-trivia system. The boundary of the system can be seen as an interface between a topoogicay non-trivia system to a topoogicay trivia system, namey the vacuum. Somewhere on the way from the center to the boundary of the system, the energy gap has to vanish, since this is required for changing the topoogica invariant from a finite integer to zero. Therefore, there wi exist ow energy states, which are bound to the region where the energy gap vanishes. From a topoogica point of view, these ow energy states must aso be very robust against externa perturbations since, as with the Chern number of the infinite system, they exist due to purey topoogica arguments. As first shown by Haperin [77], these ow energy states are ocated cose to the boundary of the system, decaying exponentiay into the buk and are therefore referred to as (topoogica) edge states. Moreover, the edge states are chira, which means they propagate in one direction ony aong a given edge of the system, and therefore are responsibe for the stricty quantized Ha conductance in finite systems. The chiraity of the edge states is aso the reason for their robustness against externa perturbations or interactions, as backscattering processes can ony occur when two or more states are counterpropagating. Without backscattering, hybridization or ocaization of the states is impossibe. Since the introduction of edge states by Haperin [77] and the introduction of the TKNN invariant for the buk system [182], many pubications have proven that the Ha conductance obtained from the infinite system via TKNN coincides with the one obtained from the anaysis of the edge states in finite system, which is known as the buk-boundary correspondence [79, 71, 44]. We wi now derive a very simpe expression for the quantized Ha conductance in terms of edge states. For this purpose, we imagine a two-dimensiona attice system on a cyindrica geometry, which consists of M y attice sites and attice spacing a y in the y-direction and M x attice sites and attice spacing a x in the x-direction. Whie the system sha be transationay invariant in the y-direction, it is confined in the x-direction 2. The confining potentia sha not be specified, the ony restriction we make for this potentia is that it sti aows for a cear distinction between buk bands and buk gaps, whie the atter may now 2 This is reaized by periodic boundary conditions in the y-direction and choosing the Landau gauge for the vector potentia A.

96 92 6. Z-Topoogica Insuators in Optica Lattices contain gapess edge states which connect the different buk bands of the system and are ocated cose to the boundary of the system, i.e. states from different boundaries have no rea space overap. Because the system is transationay invariant in the y-direction, the quasi-momentum k y remains a good quantum number, such that the eigenstates and eigenenergies of the system can be abeed as ψ α (k y ), ɛ α (k y ), respectivey, where α is an additiona abe but not a band index. Fig. 6.1 shows the integrated spectra density ρ(k y, ω) = 1 ψ α (k y ) ω H + i0 + ψ α(k y ) (6.10) α of possibe systems, where one can ceary distinguish between buk band regions and buk gap regions, which are now traversed by gapess edge states. The partices in the system are considered to be eectrons, Figure 6.1: Schematic iustration of the band structure of a finite system for different topoogica invariants N = 0, 1, 2 (from eft to right). The red region indicates a competey fied band, whie the green region indicates a empty band in between the two bands, there is a buk gap, fied with singe edge modes, which are ocated at the eft (red) and right (bue) boundary of the system. The Fermi energy is ocated in the buk gap, such that the system finds itsef in a norma insuator for N = 0, in quantum Ha state for N = 1 and in a quantum Ha state with increased Ha conductance for N = 2. The topoogica invariant N is obtained according to Eq. (6.18). with the eementary charge q = e and a Fermi energy ocated in a buk gap. Now, a votage V x is appied in the x-direction, which is sufficienty sma, such that the energy E V = ev x << Γ is much smaer than the buk gap. This votage eads to a ineary shifted chemica potentia aong the x-direction, i.e. x µ(x) = ɛ F ev x, (6.11) M x such that the difference in the chemica potentia between the eft and the right boundary of the system is µ = ev x. Since the buk of the system is gapped, the change in the chemica potentia does not infuence the buk at a. On the other hand, there are gapess states at the edges of the system, which wi change their occupation according to the changed chemica potentia. Therefore, since the chemica potentia is arger at the eft edge, there wi be more states popuated on the eft edge than on the right edge, eading to an asymmetry in the current on both sides and therefore to a net current density j y in the whoe system. This current density is determined by j y = I y = e v α (k y ) = e ɛ α (k y ), (6.12) M y a y M y a y M y a y k y α α where I y is the tota current in the y-direction, v α (k y ) = ky ɛ α (k y )/ is the veocity associated with the state ψ α (k y ) and the sum runs over a states α that are occupied on the eft side but unoccupied on the right side of the system. In the imit of a weak perturbation, this can be simpified to j y = e ɛ F (k y ) n, (6.13) M y a y k y where ɛ F (k y ) is the energy of the edge states at the Fermi surface and n is the number of additionay occupied states. The chemica potentia on the eft boundary of the system has changed by µ = ev x, due

97 6.1. The Quantum Ha Effect 93 to the appied votage. The number of additionay occupied states in the energy interva [ɛ F, ɛ F + ev x ] sha now be expressed in terms of the votage. Therefore, we rewrite the change in the chemica potentia in terms of the k-space voume k y that is avaiabe due to the energy change ev x = µ = ɛ F (k y ) k y k y. (6.14) In the ast equation, the absoute vaue of the derivative of the dispersion is taken because both the k-space voume and the energy shift are defined as being positive. The number of states in this k-space voume is straight-forward to determine. In a attice of M y sites, with attice spacing a y, the quasi-momentum k y is distributed over M y equay distributed vaues in the interva [ π/a y, π/a y ] and therefore Combining equations (6.13)-(6.15) eads to the Ha current j y = e2 ɛ F (k y ) 2π k y k y = 2π n M y a y. (6.15) ( ) ɛ F (k y ) 1 ( k y V x = e2 h sgn ɛf (k y ) k y ) V x (6.16) and therefore to the Ha conductance σ xy = j ( ) y = e2 V x h sgn ɛf (k y ) k y (6.17) which is evidenty quantized in terms of e 2 /h. This derivation for the Ha conductance was performed for the case where a singe edge state is present on the eft edge of the system. For the case of mutipe edge modes, this derivation can easiy be extended to represent the contribution of every singe edge mode to the Ha conductance as ong as they are we separated in k-space. Therefore, the combined Ha conductance for an arbitrary number of edge states is given by the sum σ xy = e2 h ( ɛα (k y ) sgn k y α ), (6.18) where α abes the edge modes crossing the Fermi edge and ɛ α (k y ) is the dispersion of the α-mode at the Fermi edge. The derivation of Eq. (6.18) presented in this thesis is very simpe and does not directy rey on any topoogica arguments, however, the presence of edge states, which is required for a non-zero vaue of σ xy, is a resut of the topoogy of the system, as argued above. However, the integer ν = α ( ) ɛα (k y ) sgn k y (6.19) can aso be interpreted in terms of topoogy, as pointed out by Hatsugai [79]. In his anaysis, he found that (6.19) is the winding number of the edge states on a Riemann surface, which is obtained by anayticay continuing the energy onto the compex pane. Additionay, Hatsugai showed that this number is identica to the tota Chern number of the occupied bands and therefore proved the buk-boundary correspondence. Whereas Eq. (6.18) is independent of the confining potentia, the formuas for the Ha conductance in [79] are derived ony for the case of open boundaries but without a trapping potentia. Whie we present a generaized formuation of the Ha conductance for finite systems (6.18), which is formay equivaent to the one obtained in [79] for the open system, no proof exists for the buk-boundary correspondence to hod for arbitrary confining potentias. For a certain confining potentia, the buk-boundary correspondence has then to be vaidated individuay by either numerica or anaytica cacuations (see section 6.3 for the case of potentias reevant in optica attice experiments).

98 94 6. Z-Topoogica Insuators in Optica Lattices 6.2 The Hofstadter Mode So far, the discussion of the quantum Ha effect was based on topoogica arguments and the existence of a band structure and did not rey on the expicit form of the underying Hamitonian. In this section, we wi discuss the famous Hofstadter mode, which was introduced by Dougas Hofstadter in 1976 [87]. In the context of the quantum Ha effect, two theoretica modes have been of great importance, the two-dimensiona eectron gas (2DEG) subjected to a perpendicuar magnetic fied and the Hofstadter mode, which in its initia formuation describes spiness fermions in a two-dimensiona attice potentia with an perpendicuar externa magnetic fied in the tight-binding approximation. The first-quantized Hamitonian for this system reads H = (p e c A)2 2m + V (x, y), (6.20) where A is the vector potentia determining the magnetic fied according to B = A and V (x, y) is the two-dimensiona attice potentia in the x-y-pane. In the foowing, we choose the Landau gauge, i.e. A = Bxe x such that B = Be z is perpendicuar to the pane. According to the origina formuation by Peiers [144], the second-quantized Hamitonian in the tight-binding approximation is given by H = t i,j e iφij c i c j, (6.21) where c j creates a partice in the Wannier state j, which is defined as the Fourier transform of the Boch eigenstates of Hamitonian (6.20) but without a magnetic fied. The so-caed Peiers phases φ ij have to be chosen such that a partice hopping on a cosed path aong the unit ce governs the phase Φ = φ 12 + φ 23 + φ 34 + φ 41 = Ba x a y, (6.22) for an iustration see Fig. 6.2 A common choice of the phases, which is often aso referred to as the Landau gauge, is { 2παxi for hopping in the y-direction φ ij =, (6.23) 0 for hopping in the x-direction where the parameter α = Ba x a y /2π is referred to as the fux through the unit ce. Figure 6.2: A partice hopping around a unit ce of the fuxess system, encosed by the attice sites 1 4, accumuates the phase Φ = 2πα according to Eq. (6.22). Modifying the phases π ij does not infuence the physics described by the Hofstadter mode as ong as Φ stays invariant under these modifications, which resuts from the gauge-freedom of the vector potentia. For α being rationa, i.e. α = p/q with p, q N, the unit ce of the system is enarged an consists of q unit ces of the fuxess system (the system without magnetic fied obviousy has α = 0), which eads to a spitting of the owest band into q so-caed magnetic bands of the system and the Briouin zone to be B = [ π/a x q, π/a x q] [ π/a y, π/a y ]. For the case of odd q, these bands are we separated by an

99 6.3. Effects of Smooth Boundaries on Topoogica Edge Modes in Optica Lattices 95 energy gap, whie for the case of even q the two bands at zero energy are touching each other at q points in the Briouin zone, eading to so-caed Dirac points in the spectrum with a inear dispersion cose to these points, however, the residua q 2 bands are again we separated from each other and from the zero energy bands by non-zero band gaps. For non-rationa fux α the spectrum shows a fracta structure, as pointed out in [87]. The spectrum of the Hofstadter mode as a function of the fux α is known as the Hofstadter butterfy and shown in Fig In the context of the quantum Ha effect, the Hofstadter mode with rationa fux is particuar interesting Figure 6.3: Energy spectrum of the Hofstadter mode in terms of the fux parameter α (6.23), which is referred to as the Hofstadter butterfy. One ceary identifies the separation of the eigenenergies into q bands for the case when α = p/q with p, q, coprime and the fracta structure for the case when α is irrationa. Figure taken from [87] with sight modification of the axis abes. because it aways shows a quantum Ha phase as soon as the chemica potentia is paced in the band gap between two distinct magnetic bands. The Hofstadter mode is particuar interesting for cod atom experiments, since it is the simpest mode on a cubic attice that reaizes topoogica phases, in this case the quantum Ha effect. In the next section, we anayze the properties of Hofstadter modes describing neutra partices in optica attice potentias subjected to artificia gauge fieds, which mimic a magnetic fied for charged partices as described in section 5.4. As we wi show, it is possibe to reaize quantum Ha physics in optica attices by experimentay reaizing the optica attice version of the Hofstadter Hamitonian. 6.3 Effects of Smooth Boundaries on Topoogica Edge Modes in Optica Lattices M. Buchhod, D. Cocks, and W. Hofstetter Phys. Rev. A 85, (2012) The foowing section discusses the effects of smooth boundaries of the edge states of a topoogica nontrivia system, which can be impemented with cod atom experiments. This paper was written in coaboration with Danie Cocks and Water Hofstetter and was pubished in Physica Review A with the tite Effects of Smooth Boundaries on Topoogica Edge Modes in Optica Lattices.

100 96 6. Z-Topoogica Insuators in Optica Lattices Since the experimenta reaization of synthetic gauge fieds for neutra atoms, the simuation of topoogicay non-trivia phases of matter with utracod atoms has become a major focus of cod atom experiments. However, severa obvious differences exist between cod atom and soid state systems, for instance the sma size of the atomic coud and the smooth confining potentia. In this chapter we show that sharp boundaries are not required to reaize quantum Ha or quantum spin Ha physics in optica attices and, on the contrary, that edge states which beong to a smooth confinement exhibit additiona interesting properties, such as spatiay resoved spitting and merging of buk bands and the emergence of robust auxiiary states in buk gaps to preserve the topoogica quantum numbers. In addition, we numericay vaidate that these states are robust against disorder. Finay, we anayze possibe detection methods, with a focus on Bragg spectroscopy, to demonstrate that the edge states can be detected and that Bragg spectroscopy can revea how topoogica edge states are connected to the different buk bands. Utracod atoms in optica attices provide a unique experimenta setup for studying properties of soid state systems in a very cean and we controed fashion [15, 29]. Particuary interesting in this context is the experimenta impementation of artificia gauge fieds for neutra atoms, as discussed in section 5.4 and in [15, 123, 124, 119, 28], simuating for instance time-reversa symmetry breaking magnetic fieds [96, 140, 167, 170, 54, 97] or a couping of the atom s interna spin degree to its anguar momentum [34, 125, 56, 120]. The reaization of these effects wi open a path for precise simuations of a arge cass of topoogicay non-trivia systems such as quantum Ha (QH) or quantum spin Ha (QSH) phases 3. Creating topoogica states of matter with cod atoms is particuary attractive because of the precise contro of physica parameters such as the hopping ampitude and interaction strength, aowing the possibiity to observe strongy interacting topoogica phases in attice experiments. However the impementation of artificia gauge fieds for neutra atoms is ony one experimenta chaenge in simuating topoogica phases in optica attices [156, 143, 2, 177]. Experiments must overcome the difficuties provided by the finite size of the attice and the soft boundary of the system, caused by a trapping potentia that is smoothy varying in space. Finite size eads to a finite overap of spatiay separated counterpropagating edge states and therefore to possibe backscattering processes, decreasing the robustness of the edge states against externa perturbations [173, 172]. Whie this is not a very serious restriction for optica attice potentias, which are reativey pure, the effects of the soft boundary of the optica attice system may significanty change the properties of the edge states characterizing topoogica insuators in finite systems. Whereas recent pubications identify the soft boundaries as an unwanted restriction or propose how to avoid them by impementing artificia sharp boundaries to their system [57], we demonstrate in this chapter that soft boundaries wi ead to interesting additiona features, either not present or at east not visibe in systems with sharp boundaries. For this purpose, we investigate different trap shapes and geometries, which are reaizabe in optica attices and discuss their specific infuence on the cod atom system. This chapter is organized in the foowing way. First, in Sec , we present the theoretica mode under consideration, a QH Hamitonian in the tight-binding approximation for spin poarized fermions confined in an additiona trapping potentia. In Sec , we present our resuts for the stripe geometry, discussing in detai the properties of the edge states in systems with a hard wa boundary, a harmonic trap and a quartic trapping potentia. In Sec , we study the shape of the edge states in a competey trapped system and investigate the suitabiity of severa detection methods as toos to probe the system experimentay, incuding Bragg spectroscopy The Hofstadter Mode for Cod Atoms The mode we consider is simiar to the ones proposed in [96, 54], experimentay reaizing time-reversa symmetry breaking topoogica insuators with utracod atomic gases. This mode describes a twodimensiona (2D) system of spin-poarized fermionic atoms subjected to a square optica attice, experiencing an artificia Abeian gauge fied A that induces an artificia uniform magnetic fied perpendicuar to the attice, B = Be z, which is simiar to the ceebrated Hofstadter Hamitonian [87] on the attice, see previous section. In our system, the gauge fied A enters the first-quantized Hamitonian of the system in form of the minima couping p p e c A, which eads to the Hamitonian: 3 For a brief introduction into the QSH effect, see the next chapter H = (p e c A)2 /2m + W (x) + V (x). (6.24)

101 6.3. Effects of Smooth Boundaries on Topoogica Edge Modes in Optica Lattices 97 Hamitonian (6.24) contains the optica attice potentia W and a spatiay dependent scaar potentia V which aows for the inhomogeneity of the attice, caused by the finite width of the aser beams creating the attice or additiona externa potentias such as a harmonic trap or an artificia hard wa boundary. For the moment we eave the detaied shape of V arbitrary, and ony assume that the non-oca matrix eements of V are negigibe (i.e. V m = δ,m V, where is the Wannier state at attice site ), which is reasonabe in our case since the potentia is either varying sowy compared to the attice spacing a, or is a step function. The second quantized form of Hamitonian (6.24) in the tight binding approximation then reads H = t c ei2πφ,m c m + V c c. (6.25),m The operator c here denotes the fermionic creation operator at attice site, with its respective annihiation operator c. The first term is the we known nearest neighbor (NN) hopping with ampitude t, and is compex due to the Peiers phases 2πφ,m that are a resut of the gauge fied, see Sec The second term corresponds to the inhomogeneity V with the oca matrix eements V V. The phases φ,m = 1 m 2π A d are not uniquey defined by the magnetic fied and depend on the gauge chosen. Here, we choose the common Landau gauge A = (0, Bx, 0), which eads to φ,m = α x (δ y,y m+1 δ y,y m 1), where x and y are the coordinates of attice site with attice spacing a = 1 and α = Φ Φ 0 represents the fux per paquette in units of the magnetic fux quantum, Φ 0 = h/e, see Eq. (6.23). Setting e = = 1, we obtain α = B 2π for the square attice. Throughout the rest of this chapter we choose the hopping t as the natura energy unit of our system. In the foowing sections we wi restrict our anaysis to the case where α = 1/6 or α = 2/5 respectivey. Our resuts for these two cases can easiy be generaized to other cases where α = p/q, with p, q N, and where topoogica edge states are predicted [80]. The experimenta reaization of a simiar mode was proposed in [57], where the authors consider a spinfu fermionic system subjected to an artificia gauge fied that simuates a magnetic fied of the form B = Bσe z, where σ = ±1 is the spin quantum number. This mode preserves time-reversa invariance and therefore aows for the reaization of QSH phases in optica attices. Because of the time-reversa symmetry, our anaysis aso appies to this mode when spin remains a good quantum number, and we wi mention the corresponding QSH phases throughout the text. A brief introduction to QSH physics can be found in the succeeding chapter. So far, we have not accounted for a Zeeman spitting due to an externa magnetic fied, a spin-orbit couping or a staggering potentia, a reaizabe in optica attices [57], see section 5.5. The physics caused by these additiona effects are indeed very interesting and eaving them out may seem quite restrictive, but the resuts we discuss in this chapter are quite genera and require ony that the states are topoogica and do not rey on the detaied nature of the edge states Edge States in Cyindrica Geometries The defining property of topoogica insuators in a semi-infinite system is the emergence of gapess edge states which are ocaized at one edge and robust against perturbations of the system, e.g. potentia or magnetic disorder. Furthermore, the presence of these states is the origin of the currents measured in QH [104, 63, 80] and QSH sampes [10, 101, 110] which are we known to be stricty quantized when S z is a good quantum number. Topoogica phases are typicay distinguished by the transport properties of the edge states, specificay by the quantized charge (or mass for neutra atoms) that is transported at a singe edge [108, 78]. One method of determining the topoogica quantum number for a given system is therefore to cacuate the energy spectrum on a cyindrica geometry and to evauate the transport properties of the edge states directy, as described in Sec Aternativey, one may determine the topoogica quantum numbers from the dispersion reation of fied bands [71] or the corresponding eigenstates [79, 182, 78] in the corresponding infinite system. In this section we wi focus on a cyindrica geometry and determine the spectra functions of the system of interest via exact diagonaization of the Hamitonian for a finite system of size We discuss the properties of the edge states of the system by anayzing the integrated spectra function in quasimomentum space and rea space for severa kinds of boundaries and show the robustness of the edge states against perturbations by switching on a disordered potentia.

102 98 6. Z-Topoogica Insuators in Optica Lattices Figure 6.4: Integrated spectra function for a system described by (6.25) with fux α = 1/6 and stripe geometry with attice sites. The stripe geometry is infinitey extended in the y-direction and as ong as transationa invariance is not broken aong this direction, the Hamitonian can be expressed in terms of k y and x. In this case, a attice corresponds to 100 attice sites in the x-direction and k y being chosen on a 100 points grid in the interva [ π/a, π/a]. Three experimentay reevant confinements of the form V (x) = V 0 (x/l) δ are shown: a) hard wa, δ, b) quartic confinement, δ = 4 and c) harmonic confinement, δ = 2. The spectra in the upper row, ρ L (k y, ω), show the k y -dependence aong the periodic direction, integrated over the eft haf of the confinement direction, see text, and the rea space spectra of the ower row, ρ(x, ω), show the x-dependence aong the confinement direction. To the right of the figure are the transport coefficients, cacuated using (6.26) with the Fermi edge set to the corresponding dotted ine. For hard wa and quartic confinement there is an appreciabe number of buk bands and the edge states are ceary distinguishabe, whereas within harmonic confinement we consider amost a of the states to be edge states. In each case we indicate eft edge states in red (gray), whie the remaining buk states are shown in back, corresponding to the eft haf of the cyinder in rea space, see ower pots. Two possibe approaches exist for designation of edge and buk states in softy confined systems. As seen in the upper row, energy regions with we defined topoogica quantum numbers can be identified in the spectrum. The corresponding states can be designated as part of the edge, and the remaining ones as the buk. Aternativey, we can define the edge as the point at which no states have energies within the range of energies covered by states at the center of the trap. We use the atter designation, athough there is itte difference between the two methods Identification of Topoogica Invariants Topoogica phases can be characterized either by anayzing the band structure properties of the infinite system, or in terms of the transport properties of the system in a confined geometry. Whie the first approach is insensitive to the specific shape of the confining potentia, the atter may in principe strongy depend on these detais. In this section we discuss the properties of edge states at an infinite wa boundary, reaized by open boundary conditions at the edges of the cyinder, henceforth referred to as stripe geometry. For this kind of boundary the topoogica quantum number of the infinite system is equivaent to the transport coefficient ν of the finite system, a reation known as the buk-boundary correspondence [79, 71]. The coefficient ν counts the difference in number of forwards-moving and backwards-moving

103 6.3. Effects of Smooth Boundaries on Topoogica Edge Modes in Optica Lattices 99 states at the Fermi edge, which represents the net transport for ow-energy excitations and hence the quantized edge current I E [80] and (6.18). Expicity we have 4 ν m = α m sign( ky ɛ αm (k y )), (6.26) where α m abes the states at the Fermi edge with energy ɛ αm (k y ) = ɛ F and m = L, R for the eft and right edge, respectivey. Eq. (6.26) can be obtained by appying the we-known Laughin argument to a cyindrica geometry and subsequenty foowing the procedure described in [79] or by foowing the our derivation in the previous chapter (6.18), whereas the atter has the advantage that no specific shape of the confining potentia has to be assumed. For the gauge A = (0, Bx, 0), the singe partice Hamitonian (6.24) obeys the symmetry H(x, p) = H( x, p), which eads to ν L = ν R. Throughout this thesis we wi ony consider the Ha transport coefficient for the eft edge ν ν L, which is identica to the topoogica Z quantum number of the infinite system and determines the Ha conductance σ xy = νe 2 /h, see Eq. (6.18). The topoogica Z 2 quantum number ν 2, which indicates QSH phases in the corresponding spin-1/2 system [57] can then be obtained, if S z is a good quantum number, by [196, 198] ν 2 = ν mod 2. (6.27) If ν 2 = 1 the system wi exhibit a QSH phase. In this section we make an expicit distinction between an edge state and an edge mode. An edge state aways refer to an eigenstate of the Hamitonian that is ocaized to one edge, whereas an edge mode refers to a series of edge states that are smoothy connected in momentum-space. Athough this distinction is not necessary for hard-wa systems, it is required for soft boundary systems Cyindrica Geometry with Open Boundary Conditions We first consider a system described by (6.25) with a step potentia V that is zero for x L x /2 and infinite esewhere, with L x sufficienty arge, providing a hard-wa boundary at the edges of the cyinder. Since the quasi-momentum in the y-direction is a we-defined quantum number and we are interested in transport coefficients for this direction, it is convenient to represent the spectrum of the system in terms of the integrated spectra density ρ L (k y, ω) 0 L/2 dxρ(x, x, k y, ω), where the spectra function is defined as ρ(x, x 1, k y, ω) = 2Im x, k y ω H + i0 + x, k y, (6.28) Figure 6.5: Integrated spectra function ρ L (k y, ω) for a system described by (6.25) with fux α = 2/5 and hard-wa (eft) and quartic (right) confinement. Edge states are shown in red (gray), whie buk states are shown in back. We integrate ony over the eft haf of the system in rea space, so as to separate the eft from the right edge states 5. In Fig. 6.4a), upper, the integrated spectra density ρ L (k y, ω) is shown for α = 1/6. One can 4 Pease note that the stripe geometry has the discrete transationa invariance of an infinite attice in y-direction and therefore the quasi-momentum k y is a we defined quantum number. 5 The integrated spectra function for the fu system can be obtained from ρ(k y, ω) = ρ L (k y, ω)+ρ R (k y, ω), where ρ R (k y, ω) is the integrated spectra function for the right haf of the system, which fufis ρ R (k y, ω) = ρ L ( k y, ω).

104 Z-Topoogica Insuators in Optica Lattices Figure 6.6: Fase coor diagram of the integrated spectra density ρ L (k y, ω) for a system described by (6.25) with fux α = 1/6, stripe geometry, and a confining potentia, V (x) = V 0 (x/l) δ. Buk bands are indicated in back and edge modes as coored curves, aso marked as (a), (b), (c), (d). There exist severa true crossings and avoided crossings in the spectra which combine to preserve the topoogica invariants for any confinement exponent δ. Auxiiary states of the corresponding edge modes are shown with dashed curves. The auxiiary states do not infuence the topoogica phases of the system, since they aways come in pairs with opposing veocities. identify the buk states which are grouped into six thick bands and the edges states, which cose the gaps between the bands. To determine the transport coefficients and possibe topoogica phases, we pace the Fermi edge in a buk gap and appy (6.26) to the dispersion of the edge states. There are severa phases visibe in this system. If the Fermi edge ies within a buk band, the system is in a trivia metaic phase. If the Fermi edge ies between the third and fourth band, there are a set of Dirac points with a inear dispersion and the phase is a semi-meta. In the buk gaps, which ie between the other bands, the edge modes pace the system in a quantum Ha phase with ν = 1 and 2 for the first and second buk gap, respectivey, and inverted for the third, fourth buk gap. In shorthand, we can specify the phases between the bands by gap 1/6 = { 1, 2, D, 2, 1}, where D represents Dirac points. In the anaogous spin-1/2 system, ν = ±1 indicates a QSH phase whereas ν = ±2 corresponds to a norma insuator, due to ack of topoogica protection. In addition we aso investigate the case where α = 2/5, see Fig. 6.5 for which we find gap 2/5 = {2, 1, 1, 2}. Note that there are no Dirac points for α = 2/5. The differences between gap 1/6 and gap 2/5 appear as different rea space behaviors within soft confining potentias that are not visibe within a hard-wa confinement, as we demonstrate in the next section Cyinder with Soft Boundaries With soft boundaries, it becomes reevant to ook at the spectra of the system in rea space aong the x axis, ρ(x, ω) = dk y ρ(x, x, k y, ω), as we as the partiay integrated spectra in quasi-momentum space aong the k y axis, ρ L (k y, ω). The quasi-momentum spectra aows us to extract transport coefficients and discuss the dispersion of the edge modes. On the other hand, the rea space spectrum exhibits some unusua features for different trapping conditions. We consider a attice of size and trapping geometries of the form V (x) = V 0 (x/l) δ where V 0 = 10t and L = 50a is chosen such that V (x = 50a) = V (x = 50a) = 10t which is arger than the energy spanned by the infinite system 8t. Three particuar vaues of δ are reevant to experiment: δ which reproduces hard-wa boundary conditions, δ = 4 for quartic confinement and δ = 2 for harmonic confinement. In most optica attice experiments, the confining potentia is a resut of the Gaussian enveope of the finite beam width of the asers and in the center of the trap we may approximate this confinement by its eading order harmonic term. However, it is has been suggested [72] that one can remove the harmonic term by superimposing an anti-trapping Gaussian beam of different detuning to the trapping beam, which then promotes the quartic term to the eading order approximation of the trapping potentia, i.e. V (x) x 4. This scheme was reaized in optica attice experiments to improve quantum phase diffusion experiments [195, 194]. We investigate these trapping geometries beow in further detai.

105 6.3. Effects of Smooth Boundaries on Topoogica Edge Modes in Optica Lattices Genera features and preservation of topoogica invariants In Fig. 6.4, we show a comparison between ρ L (k y, ω) and ρ(x, ω) for α = 1/6 and hard wa, quartic and harmonic confinements that are reevant to experiment. One can see that the potentia does not gap the system, and edge modes continue to connect the bands. The transport coefficients of the soft boundary systems, indicated to the right of Fig. 6.4, are insensitive to the trapping potentia. In other words, there exist energy ranges in which we can identify a transport coefficient, which is identica for a confinements we consider. Comparing the k y -dependent soft-boundary spectra of the upper row in Fig. 6.4, we make two observations: 1) we can readiy identify highy degenerate regions of buk bands in the hard-wa and quartic confinements, and 2) we find that the dispersion of edge modes that are present within quartic confinement do not change noticeaby when changing the confinement to the harmonic trap. In contrast, the rest of the spectrum is significanty modified, such that the ratio of buk to edge states is very sma. To define such a buk region in the soft boundary system, we assume the edge begins at a distance from the trap center where none of the states at this point overap in energy with any of the states in the very center of the trap (i.e. at x = 0). Anaogousy, we can ceary identify a buk region from the x-dependent spectra in the bottom row of Fig. 6.4 for the quartic trap, but not in the case of the harmonic trap. From this we concude that the quartic trap is ikey the best trapping potentia for observing effects of both the buk system and topoogica edge states in an experimenta setup, if it is not feasibe to artificiay impement hard wa boundaries as proposed, for exampe, in [57]. Furthermore, we observe no overap between states of different edges, which has been proposed to destroy edge states via coupings between the edges [173]. This again shows that the edge states are topoogicay protected and robust against externa changes in the potentia. In Fig. 6.5, we aso show a comparison of ρ L (k y, ω) for the case where α = 2/5 between a hard wa (eft coumn) and a quartic confined system (right coumn). We again cacuate the transport coefficients of both systems and ist these next to the pots to show that these aso coincide for a trapping potentias. To better understand the detais of the rather compicated spectra of the quartic trap, we choose to foow the edge modes of the hard-wa confinement by smoothy varying the trapping exponent δ. We show pots for δ = 60, 16, 8 and α = 1/6 in Fig. 6.6, where we have artificiay coored the spectra to indicate each edge mode. δ = 60 represents a very steep trap, and is amost identica to the hard-wa case: with the coor designation, one sees that the bue and yeow edge states are ony present in the 2nd and 3rd buk gaps, respectivey, whereas the red (marked as (a)) and green edge states (b) span two buk gaps. As the confinement is made softer, we see that an edge mode may cross the BZ more than once, and that the energy range of the edge states changes, e.g. with δ = 8 the red state (a) now extends into the 3rd buk gap. However, whenever this occurs, the state forms an avoided crossing at some higher energy with a different edge state and is forced downwards in energy, a process which preserves the vaue of the topoogica invariants. We represent this in the fase coor diagram by a dotted ine for parts of the edge states that are non-topoogica, i.e. not connecting different buk bands. For δ = 16, 8, we can consistenty see this occurring in the most energetic edge mode (coored green, (b)), which extends above the highest buk band, and forms an avoided crossing with the non-topoogica edge state created by the effect of the trapping on the highest buk band. Note that, due to the trapping potentia, severa edge states that beong to the same edge mode may exist for one vaue of energy. 2. Merging and spitting of edge states When the number of edge modes changes, as the Fermi edge crosses a buk band in the hard-wa boundary system, either an edge mode must be created, or an edge mode must merge into either the buk band itsef or with another edge mode. In the soft-boundary system we can see some very non-trivia behavior that shows the compexity of these processes. We first focus on the rea space spectra of the α = 2/5 fux system under quartic confinement, see Fig In the owest gap, we see that two different edge modes, which evove between the first and second buk

106 Z-Topoogica Insuators in Optica Lattices band, merge into a singe edge mode, which evoves between the second and third buk band. In the hard wa system, this mode is ocaized to a singe site in the x-direction and can ony be observed in quasi-momentum space. In the quartic trap, the edge states eaving the first buk band foow the shape of the quartic potentia and one may expect the same for the states eaving the second buk band. As one sees in Fig. 6.7, this is not the case. The states eaving the second buk band immediatey start to merge with the edge states from the first buk band and the resut is ony a singe mode at each edge, evoving between the second and third buk band. Athough it is not possibe to determine topoogica invariants from rea-space spectra, we can ink this merging behavior to the k y -space spectra of Fig. 6.5 and see that it eads to the correct topoogica quantum number ν = 1. The same effect is again observabe between the fourth and fifth buk band. Interestingy, the merging of these modes does not take pace via a simpe overap of the states, but a gap in rea space with negigibe spectra weight exists between the states originating from the bands and the newy-formed edge mode. In the α = 1/6 fux system, we aso see the opposite effect: the spitting of a singe buk band, to connect edge modes of different bands which are energeticay we separated. In Fig. 6.4, the integrated spectra density ρ(x, ω) shows that the modes eaving the second and fifth buk bands each spit into two curves, where a singe eigenstate has arge ampitudes on two spatiay separated attice sites. We interpret this spitting as a process that faciitates the connection between different bands which we observe in Fig For exampe, the outer part of the mode eaving the second buk band can be seen to merge at higher energies with the mode that is a product of the third and fourth bands. This connection between the bands is anaogous to the avoided crossings that we observe in the k y -dependent spectra in Fig This very non-trivia behavior of modes within the outer region of the system, combined with transport coefficients which are identica to the topoogica quantum numbers, given by the transport coefficients of the infinite system, indicates that the soft edge states are of topoogica origin. To further verify this, we address in section the robustness of these states against perturbations in terms of a disordered background potentia. 3. Reation of Edge States and Buk Bands When we ook more cosey at the dispersion of the edge modes, we can see an interesting connection to the buk bands of the system. We focus on the quasi-momentum spectra for the case α = 1/6 shown in Fig. 6.6 for increasing confinement exponent δ. The dispersion of an edge mode eaving a given buk band can be described on two different quasimomentum scaes. For a sma range of k y, the dispersion mimics that of its associated buk band, and this behavior becomes more prominent for smaer δ. This can be seen for the owest edge modes, coored red (a) and bue (b), e.g. the red (a) mode has a ocay fat dispersion, mirroring the fatness of the owest band. However, when avoided crossings have occurred, such as for the yeow (c) and green (d) edge modes, the dispersion of an edge mode cannot simpy be described by one band aone and corresponds to a mixture of bands. On the other hand, considering the Briouin zone as a whoe, the edge modes become more fat in momentum space, the smoother the confining potentia is in rea space. This fattening is a direct resut of the number of accessibe sites at the edge. The number of attice sites, n edge, that are avaiabe for an edge state between e.g. the first and the second buk band, is the number of sites i that fufi ɛ 1 ɛ 0 V (x i ) ɛ 2 ɛ 0, where ɛ 1 (ɛ 2 ) are the maximum (minimum) energies of the first and second buk band, respectivey, and ɛ 0 is the minimum energy of the first buk band [172]. In the hard wa system, n edge = 1 but becomes arger the smoother the confining potentia becomes in rea space. An interesting resut for the α = 1/6 fux per paquette, is that the fatter the potentia becomes, the fatter the owest gap edge modes become, with a corresponding increase of the effective mass of system s excitations: ( m 2 1 ky 2 ɛ(k y )). (6.29) This is generay true for soft confinements, as pointed out in [172], but the edge state structure in the case α = 1/6 aows this feature even for reativey steep potentias.

107 6.3. Effects of Smooth Boundaries on Topoogica Edge Modes in Optica Lattices 103 Figure 6.7: Integrated spectra density ρ(x, ω) for a system described by Eq on a attice with quartic confinement V. In the eft figure, the fux is chosen to be α = 2/5, whie in the right figure α = 1/6. In the eft figure, the edge states eaving the second buk band immediatey merge with the edge states of the first buk band to form a state spatiay ocaized between the buk bands with the correct transport coefficient. In the right figure, the states eaving the second buk band spit up, i.e. they ocaize to more than one point in space. The inner part of these states merges with the edge states of the third band at higher energies. Simiar behavior is observed for the 3rd, 4th and 5th band. As pointed out in the text, this non-trivia behavior is an indication of the topoogica origin of the edge states Robustness of Soft Edge States in Stripe Geometries One of the most important properties of topoogica edge states is their robustness against even arge perturbations, which eads to ceary detectabe quantized Ha conductance in impure experimenta setups. In optica attice experiments which are, by construction, very cean reaizations of condensed matter Hamitonians, perturbations such as disorder are usuay not an issue. However, since disordered potentias can be impemented in a controed manner [12, 46, 157], it is of interest to thoroughy investigate how robust the edge states are against these kinds of perturbation. Here, we address this question for soft boundaries. The genera argument, which iustrates the robustness of edge states in condensed matter systems, is the ack of possibe backscattering processes [78]. Counter-propagating edge states are ocaized on opposite edges of the system and are very we separated spatiay. Therefore, in huge condensed matter systems, these states have vanishing spatia overap and backscattering from impurities is competey suppressed. In contrast, in finite systems different edge states from opposite edges wi have a finite overap in rea space, which theoreticay aows for backscattering processes, and therefore disorder may ead to the opening of a gap in the spectrum. However, as we wi see from our numerica resuts, even in very sma systems ( 60 attice sites in x-direction) this effect is not observabe. To verify the robustness of the soft edge states numericay, we perturb the system described by Eq by adding a disordered background potentia V disorder = c c, (6.30) where is distributed randomy, either by a binary distribution, {0, max },, or by a uniform distribution, [0, max ], 6. For a reaizations, we found that the edge states stay robust and sti connect the different buk band regions without opening a gap up to disorder strengths of about max 0.5t for binary disorder and even arger strengths for uniform disorder. For exampe, in Fig. 6.8 the integrated partia spectra density ρ L (k y, ω) is shown for a attice system with uniformy distributed and max = 0.5t. There is ceary no gap in the spectrum and 6 In the presence of a disordered background potentia, transationa invariance in the y-direction is broken and therefore, to mimic a semi-infinite system, we diagonaize the Hamitonian of the system on a L x L y attice with periodic boundary conditions in the y-direction (i.e. ψ(x, y) = ψ(x, y + L y), where ψ(x, y) is the singe-partice wave function).

108 Z-Topoogica Insuators in Optica Lattices Figure 6.8: Integrated partia spectra density ρ L (k x, ω) of a system described by Eq on a attice with α = 1/6. Right: with an additiona binary disordered potentia, given by Eq and max = 0.5t. Left: with the disordered potentia being set to zero. The robust edge states are sti ceary pronounced and gapess, whie the former buk bands are smeared out and show a mobiity gap (not shown here but obtainabe from the Anderson-ocaized buk eigenstates). athough quasi-momentum is no onger a good quantum number, the edge states in momentum space are very sharpy centered around a particuar vaue of k y and remain deocaized in the y-direction as they were for the system without disorder 7. In contrast, some of the buk states now consist of many quasi-momentum components (not shown in our figure) and therefore become ocaized to a region much smaer than the system size, which can be termed Anderson ocaization. We have aso addressed arger systems with arger boundary regions. These systems contain more and more edge states in a given buk gap, which may possiby ead to different backscattering processes between edge states ocated at the same edge and therefore open gaps in the spectrum after disorder is introduced. To excude these possibiities, we studied system sizes of up to attice sites without finding any indication of gaps in the spectrum or ocaization of the edge states up to disorder strengths of max = 0.5t Detection Methods So far we focused on a semi-infinite system with stripe geometry. However, reaistic systems in optica attice experiments are confined to a finite region in a dimensions by the finite beam width of the asers. In this section, we determine the spatia wave functions of a 2D system trapped in both the x- and y- directions and discuss possibe detection methods of the resuting edge states Eigenstates of the Competey Trapped System We determine the eigenstates of a system with a confining potentia V that varies in the x- and y- directions [ ( x ) δ ( ] y δ V (x, y) = V 0 +. (6.31) L L) The parameter δ determines the shape of the trap and the possibe eigenstates. For δ the system is again confined by hard was in both directions, whie for δ = 4 and δ = 2 the system is in a quartic and harmonic confinement, respectivey. For harmonic confinement, we expect the eigenstates that are extended over various attice sites to be circuary symmetric, whereas in the quartic case the potentia is no onger circuary symmetric and the states take on a shape that is sometimes referred to as a squirce 7 At this point, one has to keep in mind that we are investigating the properties of a finite system, which has per construction ony ocaized eigenstates. Locaization or deocaization therefore means confinement to few attice sites or an extension over various sites and must be understood from the context.

109 6.3. Effects of Smooth Boundaries on Topoogica Edge Modes in Optica Lattices 105 Figure 6.9: Wave function Ψ(x, y) 2 as a function of the attice spacing a of different eigenstates of the system within compete quartic confinement. The rea space spectra density shown to the eft is a cross section of the compete system: ρ(x, y = 0, ω). Three particuar eigenstates have been shown, and their energies indicated by arrows to the spectra density. The wave functions beong to a) a buk region and b) and c) to a pair of edge states spitting up after eaving the second buk band. [128]. For this anaysis we again wi restrict ourseves to a system with attice sites and a trapping potentia with a minimum vaue of V0 = 10t aong each edge of the attice. We again focus on α = 1/6. In Fig. 6.9, we ceary see the different rea space distribution of edge states compared to buk states. The buk states are deocaized over a region of about attice sites, whie the edge states for a given energy foow the isoines of the quartic potentia, and are strongy confined to these regions. Comparing figures b) and c) for the competey trapped system one can see the spitting of the edge states eaving the second buk band as the two states have a weak overap with one another. The shape of the edge states in the quartic confinement ooks simiar to that which one woud expect in the hard wa system (ike those expicity shown in [57]) and differs ony at the very corners of the system. We therefore expect simiar singe partice excitations for the quartic confinement as for the hard wa confinement when probing the edge states in experiment. The situation sighty changes when ooking at the harmonicay confined system. There, the confining potentia is circuary symmetric and one may expect that the eigenstates refect this symmetry. The wave functions of the harmonicay confined system are shown in Fig As aready seen from the spectra density potted in Fig. 6.4, the former buk region is tighty confined to very few attice sites in the center of the trap, which makes it difficut to define a buk region in the harmonic trap. On the other hand, the edge states chosen refect the radia symmetry of the trapping potentia and again are ocaized aong the isoines of the trapping potentia. This aready indicates that for the harmonic confinement we expect very different excitation dynamics than for the quartic and hard wa confinement, where significant parts of the eigenstates are quasi-one-dimensiona. For harmonic confinement, one can sove the continuum mode anayticay in the absence of the attice [131, 27] and the resuting wave functions are quite simiar to those from the attice cacuation. The major difference in the continuum case is that no edge states from different buk bands merge, since the

110 Z-Topoogica Insuators in Optica Lattices Figure 6.10: As in figure 6.9 but with compete harmonic confinement. The three states are shown from a) a buk region and b) and c) edge states of states beonging to different buk bands. Spatia coordinates in units of the attice spacing a. ha conductivity aways increases by one when passing a buk band and the different buk bands are not connected. Additionay, anguar momentum is ony a good quantum number in the continuum case Bragg Spectroscopy An important question concerning topoogica non-trivia phases in utracod atoms is whether the edge states are detectabe with existing experimenta toos. Due to the ack of stationary transport in optica attice experiments, it is not feasibe to directy measure the Ha conductance, and one has to consider aternative approaches [172, 158]. Severa possibiities for detecting edge states or topoogica quantum numbers in optica attice experiments have been proposed. Some require carefu experimenta impementation such as Bragg [126, 58] or Raman spectroscopy [35] and others take advantage of easiy accessibe observabes ike time-of-fight (ToF) patterns or density profies. Density profie measurements were proposed by Umucaiar et a. [186] to directy separate the buk and edge densities between different bands. However, as aready pointed out in [172], these profies do not show the required structure, as can be seen in Fig. 6.4 (ower): the buk bands a occupy approximatey the same rea space extent. Hence, this method is not appicabe to topoogica systems in genera. Aternativey, ToF measurements have been proposed by Zhao et a. [200] to exhibit minima and maxima that depend on the topoogica number of the system. Whie this is true for the specific cases they were investigating and aso for our system in the case of α = 1/6, we found that it is not vaid in the case of α = 2/5 and therefore cannot be reiaby used as a detection method in experiment. In contrast, Aba et a. [3] propose using ToF measurements as a method to identify skyrmions, by focusing on topoogica properties of pseudo-spin vectors within the Hamitonian on the Boch sphere. However, this method focuses on buk properties rather than the edge modes that we consider here. We choose to focus instead on Bragg spectroscopy, which probes the dynamica structure factor S(q, ω) of the underying system. Bragg scattering of topoogica insuators in optica attices has been previousy considered for the case of the quantum anomaous Ha effect [126]. However, no inhomogeneity of the

111 6.3. Effects of Smooth Boundaries on Topoogica Edge Modes in Optica Lattices 107 attice was considered. Recenty, Godman et a. [58] have investigated Bragg spectroscopy theoreticay, considering shaped asers to probe anguar momentum states within circuary symmetric traps. Whie this is a nove impementation to enhance the detection of edge states, we demonstrate that one is abe to observe edge states using a simpe inear Bragg couping which, due to technica imitations, may be the ony option avaiabe to a particuar experiment. Furthermore one can observe differences in Bragg spectroscopy between the various bands that we show is not due to chiraity considerations. We do not propose an expicit experimenta setup and simpy assume that one can measure the dynamica structure factor directy. One such proposa to measure this precisey in an optica attice is the so caed "sheving method" [58]. When performing Bragg spectroscopy, the system is iuminated by two aser beams, described by wave vectors p 1, p 2 and frequencies ω 1 = p 1 c, ω 2 = p 2 c, respectivey, and the differences in these quantities, q = p 1 p 2 and ω = ω 1 ω 2, aows for transitions between different eigenstates of the origina system. The Hamitonian describing the interaction of the system with the aser beams is then given by H Bragg = Γ d 2 p ( e iωt Ψ (q + p)ψ(p) + h.c. ), (6.32) 2 where Ψ (p) is a fied operator, creating a partice with rea momentum p and Γ is the couping strength of the asers [171, 175, 174]. The dynamica structure factor in inear response theory for an infinite homogeneous system is directy connected to the density-density correation function χ q,q (ω) S(q, ω) = 1 π Imχ q,q(ω) (6.33) via the fuctuation dissipation theorem [142]. For our case, we have to evauate S(q, ω) for the inhomogeneous system, where the quasi-momentum is no onger a good quantum number. Within the inear response approximation, and accounting for the finite size of the system and finite time of the measurement process, we find: S(q, ω) = Γ 2 n λ (1 n µ ) A λ,µ (q) 2 (ω ω µ + ω λ ) (6.34) µ,λ Here, λ, µ abe the singe-partice eigenstates of the system, n ν and ω ν are the occupation number and energy, respectivey, of the state ν. We introduce a Lorentzian broadening factor, to aow evauation in a system of finite size. The scattering ampitude A λ,µ (q) is the probabiity of a partice in state µ to scatter into the state λ by gaining momentum q and is given by the integra A λ,µ (q) = d 3 re iqr ψµ(r)ψ λ (r). (6.35) After determining the singe partice eigenstates of the system, we can directy cacuate the dynamica structure factor. Because we are focusing on the detection of edge states, we investigate a system with a Fermi energy ocated in a buk gap at ɛ F = 2t (see Fig. 6.4), where an edge state is ocated. There are now four genera scattering processes possibe, edge edge, edge buk, buk edge and buk buk. Edge edge scattering is ceary distinguishabe by anayzing the dynamica structure factor. Given a frequency ω, the set of possibe momentum transfers aowed to another edge state is very imited because the edge states are we ocaized in momentum space. For the case of edge buk scattering, many different momenta are accessibe and therefore we see a signa regardess of the vaue of q. This means for a fixed momentum transfer q, S(q, ω) as a function of ω consists of a δ-peak approximatey around ω = qv 8 F and a smeared out region, where the buk bands are ocated. This can be seen in Fig (eft), where the first peak indicates edge edge scattering and the second and third peaks correspond to edge buk and buk buk scattering. On the other hand, for a fixed frequency ω, the response in momentum space describing edge edge scattering ooks quite different from that obtained from edge buk scattering, as one can see from Fig a) and b), respectivey. For the quartic confinement, the edge states form squirces in rea space (Fig. 6.9), which means that ow energy excitations are most favorabe in x- or y-direction, resuting in the square-ike structure of S(q, ω) in Fig. 6.11, which is approximatey described 8 In this case we used the Fermi veocity v F ky ɛ F (k y) of the edge states of the cyindrica geometry, where k y is a good quantum number, to approximate the frequency ω, which is reasonabe when ooking at Fig. 6.9.

112 Z-Topoogica Insuators in Optica Lattices Figure 6.11: Dynamica structure factor for a system with quartic confinement and Fermi energy in the first buk gap (ɛ F = 2t). Left: S(q, ω) for fixed momentum q as a function of ω. The first peak beongs to edge edge scattering and its position is sensitive to q and can be written, for sma ω, as ω q = v F,edge q. The second and third peak beong to edge buk scattering from edge states into the third and fourth buk bands, ocated around ɛ = 0 and to buk buk scattering from the first to second buk bands, where the frequency is independent of q. No signa appears of scattering from edge states to the second buk band, ocated at ω = 0.5t, indicating a disconnection between these states, i.e. these states have vanishing matrix eements of the Bragg operator. Right: S(q, ω) for fixed frequency as a function of momentum transfer q for a) edge edge scattering at ω = 0.2t, b) buk buk scattering processes at ω = 1.5t. by {q aowed } = {(q x, q y ) max{ q x, q y } q 0 = v F /ω}. In contrast, the dynamica structure factor of buk buk scattering from the first to the second band is smeared out and depends on the Fermi surface of the occupied eve at ɛ F and the Fermi surface of the unoccupied band ɛ F = ɛ F + ω. The aowed momenta are approximatey described by {q aowed } = {(q x, q y ) q x + q y ɛ F /ṽ F } and form a rough square which is rotated by ϕ = π/4 compared to the edge edge scattering. Note that as a resut of the structure of A µ,λ (q), where a minima spatia overap of the two spatia wave functions is needed for obtaining a finite scattering ampitude, high frequency edge edge scattering is exponentiay suppressed because the presence of the trap causes energeticay separated edge states to be ocaized to different distances from the center of the trap. This does not occur in the equivaent hard-wa system. For the harmonicay confined system, S(q, ω) as a function of ω for fixed q is quaitativey the same as in the quartic system. In contrast, S(q, ω) as a function of q for fixed ω for edge edge scattering ooks quite different than for quartic confinement. As seen in Fig. 6.10, the edge states have a circuar symmetry and therefore no momentum transfer direction is preferred, which eads to the set of aowed states forming a circe {q aowed } = {(q x, q y ) qx 2 + qy 2 ω/v F }, shown in Fig An important discovery of our cacuations is that there is an obvious absence of spectra weight at frequencies where we expect signas of edge buk scattering. To highight this, we cacuate an artificia Bragg response where we aow ony initia states in the energy range 2.5t < ɛ < 2t for transitions to states of higher energy. This means any signa due to possibe buk edge or buk buk transitions is suppressed. The spectra shown in Fig demonstrate edge edge signa for the first buk gap, edge buk signa to the third and fourth buk bands, but conspicuousy absent signa for the edge 2nd buk band transitions, which woud be expected for Bragg frequencies 0.5 < ω < 1.5. This impies that the first edge and second edge/buk are disconnected, i.e. have a vanishing matrix eement of the Bragg

113 6.3. Effects of Smooth Boundaries on Topoogica Edge Modes in Optica Lattices 109 Figure 6.12: S(q, ω) for a fixed frequency ω = 0.2t as a function of momentum transfer q, for a Fermi energy ɛ F = 2t. Left: S(q, ω) for the quartic confined system. The system shows a strong response when one component of q = (q x, q y ) has an absoute vaue q x,y = q 0 = ω/v F because excitations aong the x-axis, y-axis are most favorabe (see Fig. 6.9). Right: S(q, ω) for the harmonicay confined system. Here, the response is cose to circuary symmetric in q-space, refecting the shape of the eigenstates. No particuar direction is anymore favorabe, as ong as the absoute vaue of q = q 0 is fixed. Figure 6.13: Dynamica structure factor S(q, ω) for fixed momentum q as a function of frequency, for a Fermi energy ɛ F = 2t, as seen in Fig but with artificiay suppressed buk buk scattering processes. The first peak beongs to edge to edge scattering processes and is sensitive to the momentum transfer q with approximate frequency ω q = v F,edge q. The broadened peaks around ω = 1.5t beong to edge buk scattering to the third and fourth buk bands. It is ceary visibe that there is no scattering from the edge states to the second buk band, which is ocated at ω = 0.5t. operator. It is possibe to predict this behavior from the dispersion of the edge states (see Fig. 6.6), as one can see that the owest edge mode, coored in red, passes unimpeded through the second buk band, and never dispays an avoided crossing with the bue edge mode of the second band or the second band itsef, whie it aways merges with the 3rd or 4th band (with which we find non-vanishing matrix eements of the Bragg operator). For higher energies, and strong confinements, we see the opposite behavior of avoided crossings between red (a) edge modes and yeow (c) edge modes, indicating that one can expect a finite Bragg response from transitions between these states. Note that the ack of edge buk scattering is not a resut of the soft-boundaries inhibiting rea-space overap. We have performed equivaent hardwa boundary cacuations where rea-space overap is guaranteed but we again observe an absence of signa for disconnected edge buk transitions. Note aso, that we do not expect to observe a cear signa for arge frequency edge edge transitions regardess of the type of trap, as there is arger range of states

114 Z-Topoogica Insuators in Optica Lattices beneath the Fermi-edge that can be accessed with the Bragg aser. Hence, many different vaues of k y wi contribute, eading to a burred signa. In this section, we anayzed the properties of 2D topoogica edge states in softy confined systems with a confinement in one direction of the form V (x) = V 0 (x/l) δ. By varying the confining potentia from a hard-wa to a quartic or harmonic potentia, we showed that the topoogica properties of the edge states in specific buk gaps do not depend on the steepness of the confining potentia, whie a confinement sharper than harmonic is required to achieve an appreciabe buk region of the attice. We suggest that quartic confinement is suitabe to observe both edge state and buk properties, which may be reaized by superimposing attractive and repusive Gaussian beams. Furthermore, we observed the emergence of robust auxiiary edge states, which provide additiona structure to edge modes but do not infuence the topoogica quantum numbers. The main feature of these auxiiary states is that they connect edge states which are spatiay separated to buk bands of the system. This provides a mechanism to preserve topoogica invariance, as soon as the edge states and buk bands become spatiay separated. In these cases the band-structure exhibits a series of avoided crossings that act to preserve the topoogica invariant. An anaysis of the spectra density of softy confined systems in rea space reveaed the spitting and merging of edge states from different buk bands, which is aso indicative of their topoogica nature. We aso determined the wave functions of eigenstates in a competey trapped system and showed how these depend on the confining potentia. With these, we cacuated the dynamica structure factor which can, for instance, be measured by Bragg spectroscopy. We found that the dynamica structure factor can revea the edge and buk states of the system and their overap. In summary, we demonstrated that topoogica properties in utracod atomic systems with artificia gauge fieds are not sensitive to the trapping potentias avaiabe in optica attice experiments and that the edge states of these systems can be ceary detected via Bragg spectroscopy. We beieve that soft boundaries provide more detaied insight into the behavior of edge states, which cannot be observed in hard-wa systems, and are therefore worth investigation in their own right.

115 7. Z 2 -Topoogica Insuators with Interacting Utracod Fermions 7.1 Time-Reversa Invariant Topoogica Insuators In the previous chapter, we have discussed quantum Ha systems as prototypes for a Z topoogica cassification. The topoogica invariant corresponding to these systems is the Z number C, which is the Chern number of the occupied buk bands. It can equivaenty be obtained from to the buk-boundary correspondence of the edge states in a finite system. This topoogica invariant determines the Ha conductance σ xy. By appying the time-reversa operator T to a Hamitonian with a magnetic fied, one effectivey changes the sign of the magnetic fied. This aso changes the sign of the Ha conductance, i.e. the sign of the topoogica invariant is changed. This means that systems, which are time-reversa invariant, can ony have a Z topoogica quantum number of zero, i.e. are topoogicay trivia [78]. However, Kane and Mee in their semina paper [101] have found a different cass of topoogicay non-trivia insuators for the case when time-reversa symmetry is unbroken, but the non-trivia topoogy is caused by spin-orbit interactions. Crucia for this new Z 2 cassification is the presence of two eectronic favors, which we ca spin-up and spin-down. Whie the quantum Ha effect coud be expained by a singe fermionic favor, this is not possibe for a Z 2 topoogica phase. The key idea is, that time-reversa symmetry is broken for each spin aone, but not for the combination of both spins. For spin-1/2 partices, the T operator is antiunitary, i.e. T 2 = 1, since a rotation of the spin-state around the ange φ = 2π recovers the state. This directy eads to Kramer s theorem, which states that a eigenstates of a time-reversa invariant Hamitonian H are at east two fod degenerate. Kramer s theorem can be proven by contradiction: Suppose Ψ is a non-degenerate eigenstate of the time-reversa invariant Hamitonian H, then T Ψ must aso be an eigenstate corresponding to the same eigenenergy. Since Ψ is non-degenerate, T Ψ = e iϕ Ψ for some compex phase factor e iϕ. This woud mean T 2 Ψ = e iϕ 2 Ψ = Ψ, which is not aowed because 1 1. Therefore no non-degenerate states exist in a time-reversay invariant system [78]. The TKNN invariant for a T invariant system is aways C = 0, but there must be an additiona topoogica invariant for theses systems, since one can distinguish at east two fundamentay different casses of time-reversa invariant Hamitonians, as we see in the foowing. We can understand this in a very simpe picture by ooking at a quantum Ha system: consider a quantum Ha system with non-zero topoogica quantum number. The spectrum of this system has edge states traversing the buk gap, with opposite veocities on opposite edges. The T operator maps k k, such that T HT = dk T k h(k) k T = dk k h(k) k H, (7.1) B where h(k) is the subbock of H corresponding to the quasi-momentum k and the integra runs over the first Briouin zone B. Since H describes a topoogicay non-trivia system, there must exist at east one B

116 Z 2 -Topoogica Insuators with Interacting Utracod Fermions pair of subbocks h(k) and h( k), such that h(k) h( k). In contrast to the spiness system, for the spinfu system the T operator maps ( σ, k) (σ, k). Consider now a system, which is described by the Hamitonian H = H + H = (, k h(k), k +, k h( k), k ) dk. (7.2) B Appying the T operator to this Hamitonian then eads to T HT = (T, k h(k), k T + T, k h( k), k T ) dk = H, (7.3) B such that the system is trivia with respect to the Z cassification. We reaize two important consequences from equations (7.2) and (7.3). First, the system described by the Hamitonian H is time-reversay invariant, which means that the Hamitonian possiby reies ony on intrinsic effects of the system, but not on externa perturbations, such as magnetic fieds in the quantum Ha effect. Second, athough the Ha conductance of this system is aways zero for fied bands, the spin Ha conductance σxy S σxy σxy is stricty quantized and therefore there must exist another topoogica quantum number, which indicates this quantization but is different from the Z quantum number for the QH effect. The fact that this new cassification is Z 2 rather than Z, i.e. that there are ony two possibe topoogica casses of time-reversa invariant Hamitonians was rigorousy proven by Kane and Mee [101, 100]. Here, we wi focus on an expanation invoving edge states and Kramer s theorem. Consider a Hamitonian H, which is time-reversay invariant and describes a finite system. Either this Hamitonian has no edge states in the spectrum, then H is topoogicay trivia. Aternativey, there are edge states in the spectrum, such that H is possiby topoogicay non-trivia. Again we ony focus on the eft edge of the system and appy Kramer s theorem, which tes us that for any state x, k in the spectrum, there exists an energeticay degenerate state x, k. This can be seen, since T maps x, k x, k, where x is a combination of the residua quantum number characterizing the eigenstates of H and may be changed by appication of T. Therefore, it is sufficient to anayze the spectrum on the eft edge of the system and for the right haf of the first Briouin zone k y [0, π/a]. If there is one edge state present in this part of the spectrum, it must be robust against perturbations that respect time-reversa symmetry and therefore respect Kramer s theorem. If there are two edge states present in this part of the spectrum, an externa perturbation (in form of disorder or interactions) can easiy ead to a hybridization of these states and to the formation of a gap in the spectrum, which woud make the system topoogicay trivia. The hybridization of edge states is aways possibe when there is an even number of edge states present in this part of the spectrum, whie for an odd number of states there is aways one edge state protected by Kramer s theorem. From this argument, we can deduce that there indeed exist ony two distinct topoogica cassifications for timereversay invariant systems: namey topoogicay trivia (ν = 0) and topoogicay non-trivia (ν = 1), whereas a topoogicay non-trivia systems are equivaent. The topoogica quantum number can then be defined in terms of edge states ocated at a singe edge of the system. For a given Fermi energy E F, one simpy has to count the number Ñ of edge states in the interva k y [0, π/a] which cross the Fermi energy, such that ν = Ñmod2, (7.4) is the topoogica invariant for this system. In some iterature, it is sometimes conventiona to consider k y [ π/a, π/a], such that ν = Nmod4 (7.5) is the correct formua that determines the topoogica invariant, where N is the number of edge states crossing E F in the mentioned interva. The topoogica invariant for time-reversa invariant systems can aso be determined from the eigenstates of the infinite system, where no edge states are present in the spectrum [100, 50, 52, 51]. This approach is competey equivaent to the counting of edge states, since there exists a buk-boundary correspondence for Z 2 topoogica insuators as we 1 [44]. In two dimensions, the Z 2 topoogica insuator is known as a quantum spin Ha insuator 2, which was originay predicted to exist in graphene and in two dimensiona 1 The argumentation is the same as for the QH system. When there exists an interface at which the system changes its topoogy, there must be states which cose the buk gap ocated at this interface. 2 Athough theoreticay possibe, quantum spin Ha insuators do generay not have a quantized spin Ha conductivity [101].

117 7.2. Time-Reversa Invariant Hofstadter-Hubbard Mode with Utracod Fermions 113 semiconductor system with a uniform strain gradient. However, the quantum spin Ha insuator so far has ony been experimentay observed in HgCdTe quantum we structures [110, 9]. In the foowing section, we wi discuss a certain mode system for quantum spin Ha insuators, that can be engineered in cod-atom experiments [57], and investigate the effect of Hubbard interaction on this system. 7.2 Time-Reversa Invariant Hofstadter-Hubbard Mode with Utracod Fermions D. Cocks, P. P. Orth, S. Rache, M. Buchhod, K. Le Hur, and W. Hofstetter arxiv: , preprint The foowing section discusses the effect of interactions on a Z 2 topoogica insuator, namey the timereversa invariant Hofstadter mode with Rashba spin-orbit couping, which can be impemented in cod atom experiments [57]. This paper was written in coaboration with Danie Cocks, Peter Orth, Stephan Rache, Karyn Le Hur and Water Hofstetter and is avaiabe as a preprint with the tite Time-Reversa Invariant Hofstadter-Hubbard Mode with Utracod Fermions. My contribution to this work was the impementation of the spin-mixing CT-AUX sover for the RDMFT procedure and the performance of certain numerica cacuations. Utracod quantum gases trapped in optica attice potentias provide insight into strongy correated condensed matter systems. Exampes are the Mott-superfuid transition, the dynamics of the Hubbard mode after a quench of parameters and simuation of quantum magnetism [7, 160, 165]. The precise experimenta contro over amost a system parameters, incuding the partice-partice interaction strength, is remarkabe. Simuating more traditiona eectronic condensed matter systems is, however, compicated by the fact that cod atoms are charge neutra, such that the presence of a static magnetic fied does not ead to a Lorentz force, i.e. to the emergence of orbita magnetism, for these atoms. An experimenta breakthrough was thus the engineering of so-caed artificia gauge-fieds, which give rise to effective magnetic or eectric fieds for the neutra partices [34, 96, 124, 2, 97, 28], exempary described in Sec Remarkaby, they may even be generaized to simuate spin-orbit couping or couping to non-abeian fieds [143, 125, 177, 81, 24]. The effective eectro-magnetic fieds and coupings can be arge, i.e. the fux per paquette is of the same order as the density per attice site, which aows for the reaization of the quantum spin Ha effect (QSH) in a competey new experimenta context [57, 54, 78]. The underying idea of reaizing time-reversa invariant two-dimensiona (2D) topoogica phases with cod atoms is as simpe as it is fundamenta. Consider the (integer) quantum Ha effect (QHE) on a 2D square attice where an externa magnetic fied aong the z-direction breaks time-reversa and transationa symmetry. This system is described by the Hofstadter Hamitonian (6.21), which we described in the previous chapter. The singe partice spectrum for arbitrary magnetic fied strength having the shape of a butterfy, see Fig. 6.3 was first computed by Dougas Hofstadter [87] and is commony referred to as the Hofstadter butterfy. If the fux per paquette is a rationa number α = p/q, in units of the Dirac fux quantum Φ 0 = h/e, the system remains transationa invariant with an enarged unit ce that contains q attice sites, instead of a singe one for the fuxess system. The spectrum consists of q buk bands and in a buk gaps one finds a finite Chern number C and, correspondingy, at east C chira edge modes at a singe edge of the system 3. Interestingy, for even vaues of q, the system is a semi-meta at haf-fiing and exhibits q Dirac cones. Fig. 7.1 dispays the spectrum for the Hofstadter Hamitonian in the first magnetic Briouin zone for the case α = 1/2, ceary showing the q = 2 Dirac cones where the bands touch. To restore time-reversa symmetry, we can imagine appying a magnetic fied in the z-direction that ony coupes to the up-spins and a second fied of the same strength, but opposite direction, that ony coupes to the down-spins. We thus end up with a spinfu and time-reversay (TR) invariant version of the 3 A more detaied discussion can be found in chapter 6, here we just briefy review the essentia resuts.

118 Z2 -Topoogica Insuators with Interacting Utracod Fermions Figure 7.1: Singe-partice spectrum of the Hofstadter Hamitonian (7.6) for α = 1/2, γ = 0 in the first magnetic Briouin zone k [ π, π] [ pi/q, π/q] with q = 2. At zero energy, the two magnetic bands are touching each other via Dirac cones, making the system a semi-meta for a Fermi energy F = 0. The quasi-momenta are shown in units kg = (π, π/2). fundamenta Hofstadter probem. Remarkaby, such a scenario is feasibe using cod-atoms in artificia gauge fieds [57, 54]. Thus, the semi-metaic Dirac dispersion for even q becomes a generaization of graphene with a tunabe number of Dirac cones. Energy gaps which were crossed by a singe chira edge mode in the QHE setup, are now traversed by a heica Kramer s pair of edge states, corresponding to a Z2 topoogica insuator phase, as discussed in Sec.7.1. Note that one can use the same Gedankenexperiment to construct the Kane-Mee mode [101] from two time-reversed copies of Hadane s honeycomb mode [76]. The Kane-Mee mode with additiona Hubbard interaction has recenty been intensivey studied [169, 89, 152, 197, 117, 201], in contrast to the Hofstadter probem, for which the effect of interactions has up to now not been considered. In this section, we study the effect of interactions in the time-reversa invariant Hofstadter-Hubbard mode using rea-space dynamica mean-fied theory (RDMFT, see chapter 1). We expain our numerica resuts using anaytica arguments obtained from perturbation theory and renormaization group approaches. We consider interaction effects on both (semi-)metaic and gapped topoogica phases. Athough Z2 topoogica insuators are known to be robust against disorder [138, 161, 148], rigorous and genera resuts on the fate of topoogica insuators in the presence of Couomb or Hubbard interactions are imited [198, 118, 71]. Some three-dimensiona materias of the iridate famiy are possibe candidates for systems where strong spin-orbit couping and Couomb interactions compete [145, 102]. In two spatia dimensions, however, topoogica insuator phases have so far ony been found in HgTe/CdTe quantum wes [9, 110], where Couomb interactions seem to be negigibe Hofstadter-Hubbard Mode The non-interacting part of the Hamitonian consists of two independent copies of the Hofstadter mode (6.21), where the two distinct fermionic favors (which we refer to as spins) fee an opposite effective magnetic fied, i.e. a fied B = Bσz ez, where ez is the unit vector in z-direction and σz is the Paui matrix. These distinct copies are then couped by a Rashba-type spin-orbit couping in the x-direction, which

119 7.2. Time-Reversa Invariant Hofstadter-Hubbard Mode with Utracod Fermions 115 gives the possibiity to fip the spin of a partice that propagates in the x-direction. This Hamitonian can be reaized in cod-atom experiments (see Sec. 5.5) and states H 0 = t x,y c x+1,y ei2πγσx c x,y + c x,y+1 ei2πασzx c x,y + h.c.. (7.6) Here, c x,y = (c,x,y, c,x,y ) at attice site (x, y), σ x, σ z are Paui matrices and γ [0, 0.25], α = p/q are the spin-orbit couping strength, the magnetic fux per unit ce in units of the Dirac fux quanta, respectivey. In the foowing, we express a energies in units of t 1. The reaization of the noninteracting Hamitonian (7.6) in cod-atom experiments, was proposed by Godman et a. [57], where aso the reaizabe topoogica phases of this mode have been anayzed in terms of edge states for α = 1/6. According to the Z 2 cassification of TR-invariant systems, one finds non-topoogica (semi-) metaic and norma insuator phases and topoogica non-trivia QSH phases, depending on the parameter γ and the Fermi energy of the system. Figure 7.2: Magnetization m = n n in the Née state is potted versus interaction strength U. Shown are ines corresponding to the vaues α = 1/2 (bue), 1/4 (red), 1/6 (magenta), 1/8 (back), and 1/10 (green). Inset: Fermi veocity 2πv F (red symbos) for different α = 1/q is shown versus q. Aso U c (bue symbos) obtained within RDMFT versus q is shown. The magenta ine is a fit of v F to 1/q and the cyan ine of U c to 1/q 2. Pease note that the inset ony hods for α = 1/q with even q and the fit is not to be understood as an interpoation for arbitrary q. To incude the effect of interactions, we add a Hubbard-type oca interaction to the non-interacting Hamitonian (7.6), such that the tota Hamitonian reads H = H 0 + U x,y n,x,y n,x,y, (7.7) where the on-site interaction strength U > 0 can be tuned experimentay using Feshbach resonances and by adjusting the attice depth. We first consider the Hubbard-Hofstadter probem for genera α = p/q at haf-fiing. For q odd the system is metaic with a nested Fermi surface, and antiferromagnetic Née order occurs for infinitesimay sma interaction U = 0 + as for the ordinary square attice 4. For q even the situation is very different because the system is a semi-meta (SM) at haf-fiing. The non-interacting band structure exhibits q Dirac cones (with a mutipicity of 2 due to the spin), which are separated by momentum 2π/q in momentum space 5. The α = 1/2 case is thus very simiar to graphene (but note that the coordination number for the square attice is z = 4 rather than z = 3 for the case of graphene). For smaer α on the other hand, the 4 This is in contrast to three dimensions, where a critica interaction strength U c > 0 is required for antiferromagnetic ordering. 5 The attice constant a of the square attice is set to unity a 1.

120 Z 2 -Topoogica Insuators with Interacting Utracod Fermions system embodies a generaization of graphene with a tunabe number of vaeys. We investigate the SM-insuator transition for various α = 1/q (q even) within RDMFT. In Fig. 7.2, the magnetization is shown as a function of interaction strength U. The insuating phase for U > U c is antiferromagneticay (AF) ordered with a magnetization pointing in the z-direction and an ordering wave-vector Q = (π, π), which is the common Née vector. We find that the critica vaue of U c to enter the insuating and magneticay ordered phases decreases with increasing q. This is expected from the increasing scattering that can take pace between the cones. At U c we aso observe a simutaneous opening of a singe partice gap. Within our approach we thus find no sign of an intermediate nonmagnetic gapped phase, which woud indicate a possibe spin-iquid phase [130, 8]. To understand the behavior of U c (q), we make use of Herbut s argument [85] (see Sec. 7.3). Herbut considered graphene and studied the SM-insuator transition within a arge-n approach, and found that U c depends on 2N, the number of Dirac cones (N refers to the spin degeneracy), and the Fermi veocity v F as U c v F /2N. As shown in detai in Sec. 7.3, we are abe to match our resuts with Herbut s anaysis by repacing the Fermi veocities and 2N = qn, where q is again the number of Dirac cones in the singe-partice spectrum. In fact, from the band structure at U = 0, we find v F 1/q. Consequenty, setting N = 2 for spin-1/2 partices, U c shoud exhibit a 1/q 2 behavior, which agrees very we with the RDMFT data, shown in the inset of Fig We further note that α = 3/8, which exhibits a different v F than α = 1/8, is in agreement with our findings Tunabe Magnetic Order In this part, we consider the effect of finite γ in the Hamitonian (7.6), i.e. the presence of Rashba-type spin-orbit couping that breaks the axia spin symmetry, on the interaction induced magnetic ordering. Finite γ does change the type of magnetic order in genera. To demonstrate this, we consider fixed U = 5 at α = 1/6 and cacuate the magnetization pattern for γ = and γ = 0.25 in Fig. 7.3 obtained within RDMFT. We obtain simiar resuts for other vaues of α and γ. For γ = 0.125, the magnetization ies in the S y S z pane and the spatiay dependent magnetization reads m(r) = s tot (0, cos πx 3 cos πy, sin πx 3 cos πy), (7.8) where s tot is the moduus of the magnetization and is a function of the interaction strength U and the spatia coordinate r. For γ = 0.25, the magnetic order is given by m(r) = s tot (0, 0, cos πy). (7.9) Tuning the parameter γ, we thus pass from Née order (γ = 0) to spira order (γ = 0.125, shown in Fig. 7.3 and Eq. (7.8)) to coinear order (γ = 0.25, shown in Fig. 7.3 and Eq. (7.9)), thereby crossing two magnetic quantum phase transitions. Finay, we note that the moduus of the magnetization s tot is staggered for the intermediate vaue of U shown in Fig The staggering decreases for arger vaues of U, reducing its spatia dependence in the imit U. We can quaitativey understand this type of magnetic order by rigorousy deriving a quantum spin Hamitonian for even stronger interactions when charge fuctuations freeze out competey at haf-fiing. For the derivation, we consider a Hubbard mode with a finite number of N attice sites at haf-fiing, which reads N H = H U + H t = U c i ei ˆϕij c j, (7.10) i=1 n,i n,i t i,j where the phase ˆϕ ij is a attice site dependent 2 2 matrix, according to Eq. (7.6). In the strong couping imit, i.e. U >> t, the hopping Hamitonian H t is considered as the perturbation and the ground state of the unperturbed system is the 2 N times degenerate 6. To obtain a ow-energy effective Hamitonian, we appy a unitary transformation to the Hamitonian (7.10), which is known as the Schrieffer-Woff transformation. Consider therefore the hermitian operator S, such that H = e is He is (7.11) 6 The ground state at haf-fiing consists of N singy occupied sites, where two spin degrees of freedom are possibe at every attice site.

121 7.2. Time-Reversa Invariant Hofstadter-Hubbard Mode with Utracod Fermions 117 Figure 7.3: Rea space magnetization profie m(r) in S y S z pane for α = 1/6, U = 5 and γ = (a) and γ = 0.25, respectivey. The spira order in (a) and coinear order in (b) can be expained by the effective spin mode (7.22). The coor scheme indicates the moduus of the magnetization, which is staggered for intermediate U. is a unitary transformation of the origina Hamitonian. Expanding this transformation up to second order in the operator S eads to H = H + i [H, S] 1 2 [[H, S], S] + O(S3 ). (7.12) Choosing the operator S such that resuts for the Hamitonian in i[h U, S] = H t (7.13) H = H U + i 2 [H t, S] + O(S 3 ). (7.14) By making use of (7.13), the matrix eements of the operator S can be expressed in the eigenbasis of H U, which we ca B, eading to m S n = i m H t n E m E n, (7.15) where m, n B and E m, E n are the eigenenergies of H U for the corresponding eigenstates. The diagona eements of S can be chosen equa to zero, since H t has vanishing diagona eements in the eigenbasis of H U. The matrix representation of S can be expoited, eading to H t S = H t m m S n n = i H t m m H t n n. (7.16) E m,n m,n m E n After inserting (7.16) and its hermitian conjugate, the Hamitonian (7.14) transforms to H = H U 1 ( H t m m H t n n + n n H ) t m m H t. (7.17) 2 E m E n E m E n m,n Projecting the Hamitonian (7.17) to the manifod of degenerate ground states, where the ow energy physics takes pace, eads to the states n beonging to the ground state manifod, with E n = 0. On the other hand, the appication of H t aways takes states out of the ground state manifod, such that m must aways be an excited eigenstate with eigenenergy E m = U. We can therefore drop both sums in (7.17) and write H = H U 1 U P H2 t P, (7.18) where P is the projector onto the ground state manifod. Since the first operator on the right of Eq. (7.18) is equa to zero in the ground state manifod, ony contributions from the second operator have to be taken into account. The ony non-zero contributions to the Hamitonian can be expressed as H = t2 U c i ei ˆϕij c j c j ei ˆϕji c i. (7.19) i,j

122 Z 2 -Topoogica Insuators with Interacting Utracod Fermions This operator can now be transformed into a sum of spin-1/2 operators, resuting in a Heisenberg-type mode and recovering the antiferromagnetic Heisenberg Hamitonian for e i ˆϕij = 1. For the Hamitonian (7.6), the matrix eements for couping in the x-direction and y-direction have to be treated separatey. The couping in the y-direction is described by the matrices e i ˆϕij = e i2πxiσz, such that for a specific summand of (7.19) ( ) c i ei ˆϕij c j c j ei ˆϕji c i = c (1 n,j ) c,j c,j i ei2παxiσz c,j c e i2παxiσz c,j (1 n,j ) i ( = c S j z S j ) i ei2παxiσz e i2παxiσz c i S j + S j z ( = c Sz j S j ) e i4παxi i S+e j i4παxi Sz j c i ( = 2SzS j z i cos(4παx i ) S+S j i + S S j + i where we have defined the spin operators ) ( ) i sin(4παx i ) S+S j i S S j + i, (7.20) Sz i = n,i n,i, S+ i = Sx i + isy i = c,i 2 c,i. (7.21) For the couping in the x-direction, we find a simiar expression, which in combination with (7.20) eads to the Heisenberg-type effective spin Hamitonian H = J x,y { S x,y x Sx x+1,y + sin(4πγ) [ Sz x,y Sy x+1,y + J x,y { S x,y z Sz x,y+1 + sin(4παx) [ Sy x,y Sx x,y+1 + cos(4πγ) [ Sy x,y Sy x+1,y S x,y y Sz x+1,y ]} + cos(4παx) [ Sx x,y Sx x,y+1 S x,y x + S x,y z + S x,y y Sz x+1,y ] Sy x,y+1 ] Sy x,y+1 ]}, (7.22) where we have introduced the couping constant J 4t 2 /U. The first part of Eq. (7.22) describes a spin exchange in x-direction. For γ = n 2 with n Z, we obtain a simpe antiferromagnetic Heisenberg term. Other vaues of γ, however, break the SU(2) symmetry and cause anisotropy of XXZ-type with S x as the anisotropy direction in spin space. For γ n 4, there is an additiona Dzyaoshinskii-Moriya (DM) interaction term in the YZ-pane [43, 139], which is responsibe for the spira spin order in Fig. 7.3(a). Spin exchange in the y-direction is periodic with an extended unit ce in the x-direction depending on the fux α = p/q: for odd q the unit ce contains q attice sites, but for even q it ony contains q/2 attice sites, refecting second order perturbation theory. For instance, one finds for the π-fux attice (α = 1/2) an ordinary Heisenberg exchange term. For other vaues of α, the XY-term exhibits a moduation of its ampitude depending on α, whie the Z-term aways favors AF Ising order. This rich magnetic order predicted by the spin Hamitonian is in agreement with our RDMFT findings Topoogica Phases in the Hofstadter-Hubbard Mode In this section, we study the effects of interactions on systems that have a buk gap, but possibe gapess edge excitations, i.e. the effect of interactions on the Z 2 cassification. For U = 0, we distinguish the norma (NI) and topoogica (TI) insuating phases by cacuating the Z 2 invariant ν using Hatsugai s method [52]. For U > 0, we identify the phases by computing the spectra function in a cyindrica geometry using RDMFT and counting the number of gapess heica edge states crossing the buk gap. The TI phase exhibits an odd number of heica Kramer s pairs per edge whie the NI phase has an even number,

123 7.2. Time-Reversa Invariant Hofstadter-Hubbard Mode with Utracod Fermions 119 Figure 7.4: E F -λ-phase diagram for α = 1/6 at γ = U = 0. We find insuating phases for fiings n F = /6, with {0, 1, 2, 4, 5, 6} and (semi-)metaic phases otherwise. The insuating phases are either norma (NI) or quantum spin Ha (QSH) insuators. The staggering potentia induces a phase transition from NI to QSH for = 2, 4 and aso from semi-meta to NI at haf-fiing = 3. incuding zero. Edge states are aso crucia for detecting topoogica phases in cod-atom experiments, and we numericay study how robust they are with respect to interactions. In the foowing, we focus on fixed α = 1/6, which quaitativey captures a phenomena that occur in this system for genera α = p/q. We aso consider an additiona term in the Hamitonian, that is avaiabe in cod-atom setups [57, 54]: a staggering of the optica attice potentia aong the x-direction H λ = λ j ( 1) x c j c j, (7.23) which is added to the Hofstadter-Hubbard Hamitonian (7.7). In the axia symmetric case of γ = 0, there exist TI phases ony away from haf-fiing, since the system is a (semi-)meta for n F = 1 (and not too arge λ, U). This is shown in Fig. 7.4, and is expected as the spiness Hofstadter probem at α = 1/6 exhibits a QHE with Chern number C = ±2 if ɛ F ies within the two energy gaps cosest to zero and a QHE with C = ±1 for ɛ F in the other gaps 7 (see Fig. 7.5). The Chern number corresponds to the number of chira edge modes in an open geometry. In the time-reversa invariant system at hand, we thus find an according number of heica Kramer s pairs within the gaps. For a fiing of n F = 1/6, 5/6 the system is thus a TI. We observe this topoogica phase to be stabe even for arge interactions up to U = 10. A NI-TI phase transition can be induced in the other gap for n F = 2/6, 4/6 by appying a sufficienty arge staggered attice potentia λ 1 (see Fig. 7.4). Fixing n F = 2/6, we now turn on interactions, and observe that this phase is quite stabe as shown in Fig Eventuay, arge enough interactions reverse the effect of the staggering potentia and drive the system into the NI phase. Note that a static Hartree-ike approximation (red dashed ine) yieds comparabe resuts for sma U but overestimates the effect of staggering for arger vaues of U. A topoogica phase at haf-fiing occurs ony if we break the axia symmetry in the system by considering γ > 0. We present both the non-interacting λ-γ phase diagram 8, shown in Fig. 7.7(a), and the interacting λ U phase diagram for different vaues of γ, which is shown in Fig. 7.7(b). Both semi-meta and QSH phases are robust up to interaction strengths of order U 3 5, at which point arger interactions drive the system into a magneticay ordered state. A quaitative understanding of the interacting phase diagram foows from the observation that interactions mainy reverse the effect of staggering. Prominenty, we observe an interaction-driven NI to QSH transition for γ = 0.25 and λ 1.5, and a meta-qsh transition for 0.22 γ < 0.25 and λ 1. Using RDMFT for a cyinder geometry, we are abe to directy observe the behavior of the edge states in the interacting system. Gapess edge states are key ingredients to different detection schemes of topoogica phases in cod-atom systems [173, 172, 200, 3]. Since topoogica (QSH) phases are uniquey characterized by their heica edge states [196], a probe of 7 The pus, minus sign for the Chern number C here refers to spin-up, spin-down fermions, which fee opposite magnetic fieds and therefore the buk gaps beong a Chern number with opposite sign. 8 The non-interacting λ-γ phase diagram has aso been obtained in [57], however, they have made a sma mistake in their cacuations, such that their resut differs from the one that is presented here.

124 Z 2 -Topoogica Insuators with Interacting Utracod Fermions Figure 7.5: Spectrum of the spin-up fermions for the case α = 1/6, showing QH phases for the first and second buk gap with Chern numbers C = 1, 2, respectivey. The corresponding topoogica phase for a system described by the time-reversa invariant Hofstadter-Hubbard mode has the topoogica invariant ν = C mod 2 = 1, 0, i.e. a QSH phase for the first buk gap and a NI phase in the second buk gap. Figure 7.6: Left: U-λ-phase diagram at n F = 2/6 and γ = 0 obtained within RDMFT, the red ine indicates the phase boundary obtained by a Hartree-Fock-type static mean-fied theory, which coincides with RDMFT for sma interactions. Interactions reverse the staggering induced phase transition from a NI to a TI. This can be understood within a simpe mean-fied picture, as the interactions prefer a uniform density distribution, whie the staggering prefers a staggered density distribution. Right: Mean-fied picture of compensation of the staggering potentia by interactions. these states is the most direct measurement [57, 58, 21]. In Fig. 7.8, we give an exampe of the spectra function A(k y, ω) for the interaction driven NI-QSH transition at γ = 0.25, λ = 1.5. For U = 0.5, we find no gapess edge modes that connect the two buk bands, which correspond to the NI. On the other hand, at U = 2, we ceary find a singe pair of heica edge modes traversing the buk gap, which corresponds to the QSH phase. In this section, we investigated the Hofstadter-Hubbard mode using RDMFT compemented by anaytica arguments. We quantitativey determined the interacting phase diagram incuding two additiona terms reevant in cod-atom experiments, a attice staggering and Rashba-type spin-orbit couping. Interactions drive various phase transitions. Simiar to graphene, we find that a semi-meta at haf-fiing turns into a magnetic insuator at a critica finite interaction strength. Rashba-type spin-orbit interactions ead to tunabe magnetic order with coinear and spira phases. We expicity demonstrate the stabiity of the topoogica phases with respect to interactions, and verify the existence of robust heica edge states in the strongy correated TI phase, which is crucia for experimenta detection schemes. 7.3 Herbut s Argument In Fig. 7.2 in Sec , we presented numerica RDMFT resuts for the critica on-site interaction strength U c as a function of 1/q, which marks a zero temperature quantum phase transition between a semimetaic phase and a magneticay ordered phase. The system deveops a singe-partice gap inside the