Lattice approach to the BCS-BEC crossover in dilute systems: a MF and DMFT approach

Transcription

1 Scuola di Dottorato Vito Volterra Dottorato di Ricerca in Fisica Lattice approach to the BCS-BEC crossover in dilute systems: a MF and DMFT approach Thesis submitted to obtain the degree of Dottore di Ricerca Philosophi æ Doctor PhD in Physics XXI cycle October 2008 by Antonio Privitera Program Coordinator Prof. Enzo Marinari Thesis Advisors Dott. Massimo Capone Prof. Claudio Castellani

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3 Contents Introduction i 1 An introduction to ultracold Fermi gases A (very personal) Overview on Cold Atoms Physics Some information on ultracold Fermi gas experiments Basic theoretical tools for cold atoms physics Fano-Feshbach Resonance Optical lattices and idealized solid state hamiltonians The BCS-BEC Crossover Brief History of a Long-Standing Problem Crossover in ultracold Fermi gases Leggett theory Universality and unitary limit Quantitative theories of the crossover Attractive Hubbard model and DMFT technique Introduction The Attractive Hubbard Model General properties of the model Dynamical Mean Field Theory DMFT scenario of the BCS-BEC Crossover in the Attractive Hubbard Model 61 4 Mean-Field Approach to the dilute limit in lattice models Reaching the continuum limit from a lattice model From high density to low density BCS-BEC crossover in a lattice model Mean field theory of the attractive Hubbard model Lattice model: numerical solutions for the crossover Unitarity Compressibility, ground state energy and universal relations Lattice gas

4 ii CONTENTS 4.6 Conclusions DMFT approach to the Unitary Fermi gas Introduction Ground state properties Density and SF order parameter Ground State Energy Low-density Scaling and Universal Properties Fake Universality Self-energy scaling and dilute limit Comments and conclusions Conclusions 119 A DMFT Technicalities 123 A.1 Residual density and optimal settings A.2 Numerical consistency A.3 ns Scaling A.4 Conclusions Bibliography 152

5 Introduction In recent years cold atoms provided us with a very powerful tool to investigate many-body sistems in a very clean and controlled framework. Nowadays a very fruitful line of research is the investigation of the properties of ultracold Fermi gases, which allows for the exciting possibility of mimicking the behaviour of electrons in condensed matter. Exploiting Fano-Feshbach resonances it is possible to tune the interaction strength between these Fermi atoms with an unbelievable simplicity by applying an external magnetic field. This phenomenon provides the possibility of experimentally explore, in a systematic way and in a unique framework, the so-called BCS-BEC crossover in mixtures of fermionic atoms in different hyperfine states, which play the role of spin in electrons. Varying the coupling between fermions in two hyperfine states, a system of fermionic atoms in the superfluid phase can be driven from the weak-coupling (BCS from Bardeen-Cooper-Schrieffer) regime, where the fermionic pairs are large and strongly overlapped, to the strong-coupling (BEC from Bose-Einstein Condensation) regime where the fermions pair to form tightly-bound composite bosons. The evolution between these two limits is smooth and has been longtime studied in condensed matter, also for the possible relevance of a bosonic superconductivity scenario in cuprates High-T c superconductors. However, the quantitative understanding of the properties of the system all along the crossover is still under debate, because of the lack of a small parameter in the intermediate region. It is crucial to realize that the crossover in ultracold Fermi gas occurs in a dilute system. The dilution condition can be espressed suitably in terms of a dilution parameter k F b, where k F is the Fermi momentum which is proportional to the inverse of the interparticle distance and b is the range of the potential, as k F b 1. When a Fermi system fulfills this condition, its properties can be suitably espressed only in terms of the so-called gas parameter k F a s, where a s is the s-wave scattering length. The specific shape of the interaction enters in the problem only through the scattering length. Therefore, if one compares different dilute systems for the same value of the

6 ii Introduction gas parameter the physics of the system will be universal, i.e. indipendent on the specific system under consideration, whereas, if the system is not dilute, its features depend in general on the detailed shape of the interparticle potential. Moreover, when the coupling is fixed at the point where the twobody problem starts allowing a bound state, the scattering length diverges and there is no longer a scale related to the interaction. This condition is referred to as the unitary point and there a dilute gas shows, in addiction, remarkable universal properties. From a theoretical point of view, the dilution condition implies that any model potential leading to the right value of the scattering length a s can be used to address the properties of the system. Yet, some care has to be taken in order to be safely in a sufficiently dilute regime, where the specific features of the model are not relevant. In analytic approaches to the crossover the dilution condition can in principle be realized for every density using a contact interaction (b = 0), even though this interaction requires an ultraviolet regularization procedure. However in numerical approaches like Quantum Monte Carlo (QMC) simulations in continuum space, the zero-range contact interaction is not of practical use and the dilution condition has to be enforced reducing the density. This, in principle, can lead to (non-universal) finite-range corrections to the universal properties, if the system is not dilute enough. An alternative approach has been based on QMC simulations of lattice models with on-site V ij = Uδ ij (Hubbard) interaction. The presence of the lattice provides a natural regularization of the interaction associated with the lattice spacing l. Moreover, in QMC simulations of Fermi systems with local attractive interaction, the sign-problem, which generally affects QMC simulations of fermions, can be avoided, at least if the system is not polarized. However lattice models are intrinsically different from models in continuum space because the Galileian invariance is explicitly broken. These lattice models become equivalent to continuum models with contact interaction if the density of particles per lattice site is made vanishingly small. Yet, any practical implementation in numerical approaches deals with a finite density and, to confidently extract the universal behaviour in the dilute limit, it is necessary to have some quantitative estimate of the relevance of these finite-density corrections to the dilute limit. At the moment, to our knowledge, the problem of estimating quantitatively the relevance of these corrections as a function of the density has not been addressed in a systematic way. This problem is obviously crucial in the estrapolation from lattice approaches of the so-called universal parameters, which describe the behaviour of the dilute Fermi gas in the unitary limit ((k F a s ) 1 = 0) introduced above. Any lattice QMC approach deals unavoidably with a finitesize system and a finite number of particles. In modern implementation the numbers of sites is limited to N s and the minimum density addressable to observe many-body effects is larger than 2 N s.

7 Introduction iii The focus of this thesis is the role of finite-density corrections to the dilute limit in lattice methods. In order to properly address finite-density corrections we consider two methods which are directly implemented in the thermodynamic limit and which can access any value of density (at least in principle): static (MF) and dynamical mean-field theory (DMFT). We studied within these methods the ground state properties of lattice models with Hubbard interaction down to very low densities. We anticipate that µ finite-density corrections to the rescaled chemical potential E F and rescaled gap parameter = E F, where E F is the Fermi energy, vanish very slowly (n 3) 1 already at mean-field level, irrispective on the dispersion ǫ k of the lattice model in use. In the case of µ E F (n), we find a non-monotonic behaviour that, together with the slow asymptotyc convergence, suggests that a reliable extrapolation to the dilute limit has to be limited to a small range of densities, which at MF level is well below n 0.1. Treating more accurately the correlations in the ground state replacing MF with DMFT, the dilute limit seems to move toward much lower densities and the relevance of the corrections, at least in the intermediate density region, is strongly increased. The plan of the thesis is the following: In Chapter 1 we sketch a rapid overview of some aspects of ultracold Fermi gases. Then we discuss the key theoretical concepts to understand how the BCS-BEC crossover can be realized using a a dilute Fermi gases interacting through a Fano-Feschbach resonance. In particular we introduce the concept of scattering length a s, gas parameter k F a s and we give a very simple introduction to the physics behind the Fano-Feschbach resonance. In the last section we briefly describe how cold atoms on optical lattices can be used to simulate condensed matter hamiltonians. In Chapter 2 we review the main properties of the BCS-BEC crossover starting with an hystorical overview and then focusing more on the crossover in dilute Fermi gases. We explain then the concept of universality and the main properties of the unitary limit. In the last part of the chapter, we review the main results available in literature for the crossover with a particular emphasys on the universal parameters at unitarity. A special attention is devoted to the comparison between different QMC approaches, both in continuum space and on a lattice. In the first part of Chapter 3 we introduce the attractive Hubbard model, which we use in the Chapters 4 and 5 to address the problem of the dilute Fermi gas starting from a lattice model. We review the main known results on the model for densities n = O(1), which is the density regime typically investigated in condensed matter. In the second part of the chapter we introduce the DMFT approach and the

8 iv Introduction Exact Diagonalization (ED) algorithm we use for solve the DMFT equations. In Chapter 4 we first explain how the connection between a lattice model in the limit n 0 and a dilute system in free space is particularly simple if the lattice model is reexpressed in terms of the relevant low-density variables (k F,a s ). Therefore we reformulate the mean-field approach to the attractive Hubbard model in terms of these variables. We solve numerically the mean-field equations in the crossover and we analize the relevance of the corrections to the dilute limit, with a special emphasys on the unitary point. In the last part of the chapter we also solve a model frequently used in literature, where the dispersion ǫ k = k2 2m is equal to the free particles case up to a cut-off energy Λ and compare this model at unitarity with previous results on the Hubbard model. In Chapter 5 we address the problem of the unitary limit from a lattice perspective using the DMFT approach introduced in Chapter 3. We studied the ground state properties of the attractive Hubbard model fixing the coupling to the value where the scattering length diverges and reducing the density at fixed coupling. In the last part of the chapter, we compare our results with the available DMFT data present in literature for the same problem and with QMC approaches.

9 Chapter 1 An introduction to ultracold Fermi gases 1.1 A (very personal) Overview on Cold Atoms Physics In 1993 C. Cohen-Tannoudji, S. Chu and W. Phillips received the Nobel Prize for the development of the laser-cooling technique, which allowed to lower the temperature of a bunch of atoms using a suitably chosen laser field up to an unprecedented range. The discovery triggered a revolution in the field of atomic physics, leading to a new field of research which is labelled as the ultracold atoms physics. The first investigations were mainly focused on bosonic gases to study coherence phenomena. A milestone in this direction has been certainly the direct experimental observation in 1995 of the Bose-Einstein Condensation (BEC) predicted from A. Einstein in 1924 after reading the papers of S.Bose on the photons statistic. This result led to a second Nobel price, achieved in 2001 by E. Cornell, W. Ketterle and C.Wieman. One of the amazing properties of these systems of dilute bosons is that the existing theories perfectly described the experiments [1] and the major challenges came from the experimental side. It is indeed well know that also a gas of free Bosons shows a phase transition to a condensed phase, as a direct consequence of the Bose-Einstein statistics, when the thermal de Broglie wavelength λ db becomes comparable with the interparticle spacing 2π λ db = 2 mk B T = n 1 Thus the interaction is usually not expected to play a key role if the system is dilute and it can be eventually taken in account with a mean-field description. In last years a considerable attention has been devoted to Fermi gases, even though the bosonic side is still very active. Compared with the Bosons case, Fermi systems exhibit important basic differences which are direct

10 2 An introduction to ultracold Fermi gases Stage Temperature Density T/T F Two-species oven 720 K cm Laser cooling 1 mk cm (Zeeman slower & MOT) Sympathetic cooling 1 µk cm (Magnetic trap) Evaporative cooling 50 nk cm (Optical trap) Table 1.1: (From [2]) The various preparatory stages towards a superfluid Fermi gas in the MIT experiment. Through a combination of laser cooling, sympathetic cooling with sodium atoms, and evaporative cooling, the temperature is reduced by 10 orders of magnitude. consequences of the Pauli exclusion principle. At low-temperature, a dilute fully polarized Fermi gas is not interacting, since s-wave scattering processes, which are expected to be dominant in the dilute regime, are suppressed because of the Pauli principle. This provides a unique opportunity for an almost perfect realization of an ideal Fermi gas, which is one of the paradigm of the quantum statistical mechanics. The same property makes the thermalization of such systems very slow due to the lack of scattering processes. To overcome this limitation, the experimentalists had to develop new cooling techniques, like e.g. the so-called sympathetic cooling, and this is probably the main reason for the delay in the investigation of ultracold Fermi gases with respect to Bose gases. Nowadays cooling a system of Fermions involves a lot of different steps, summarized in Tab. 1.1, in which a part of the atoms is lost and this lead to a reduction of the number of atoms cooled and trapped available in the final stage. On the other side, in order to observe interaction effects in ultracold Fermi gases, it is necessary to cool and trap together at least two hyperfine states of the same atomic species. Cooling a free Fermi gas, the transition from Boltzmann classical statistic to the quantum degenerate regime is a crossover and shows no sign of phase transitions. Then the superfluidity 1 is a result of non-trivial many-body mechanisms where interactions play a crucial 1 Usually people talk about fermionic superconductivity. However, in this case the fermionic atoms are neutral, therefore, strictly speaking, they are superfluid and we will talk about superfluids indifferently for fermionic and bosonic systems.

11 1.1 A (very personal) Overview on Cold Atoms Physics 3 role giving rise to pairing effects. In a very naive picture, which is not completely correct, Fermions have to pair forming composite Bosons before they can condense. Then the physics of ultracold Fermi gas is richer than the corresponding bosonic counterpart and also quite harder to address quantitatively, especially in the strongly interacting regime. To cut a long story short, the field of ultracold Fermi gases started in 1999 when B. De Marco and D. Jin first succeeded in cooling Fermi atoms down to the quantum degenerate regime [3]. A key feature of these Fermi gases is that the effective interaction acting between fermions in different hyperfine states can be varied in an astonishingly easy way by applying an external magnetical field, exploiting the so-called Fano-Feshbach resonance 2. This effect was well-known from atomic physics and its use in the context of ultracold atoms was first proposed by H. Stoof in 1993 [4]. Even though, in principle, it is not restricted only to Fermi gases, for Bose gas near the resonance a sharp increase of inelastic processes has been observed, making much harder to work-out measurements before the decay [5]. It is indeed important to realize that for so low temperature, these systems should be clearly in the solid state. However the density is so low that they usually stay in a gas-like metastable phase for long enough to be observed. However, the rate of three-body processes, which are the leading processes for the formation of solids, is strongly dependent on the strength of the interaction. Thus for bosons the lifetime of the metastable state is strongly suppressed near the resonance, whereas it has been theoretically demonstrated in [6] (and experimentally observed in several experiments (see e.g. [7]) that Fermi gases near a Fano-Feshbach resonance show a remarkable increase of the collisional stability against three body losses, as a consequence of the Pauli exclusion principle. The resonance gave us the unprecedented possibility of exploring different regimes of a Fermi system in a uniquely controlled framework. This is in clear contrast with exploring different coupling regimes in condensed matter systems, where, in the best case, it is necessary to apply some pressure or varying the chemical composition doping the system. This usually involves many side-effects eventually forbidding a very easy understanding of what is really happening in the system. The introduction of a method to control the coupling triggered the development of several different lines of reseach. One of the main thread is certainly exploring the possible applications of cold Fermi atoms in the realization of system of condensed matter system. This line of research, which has its final aim in the production of a sort of Quantum Simulator for paradigmatic models in condensed matter, like the Hubbard model which A basic introduction to the physics of Fano-Feshbach resonance will be given in Sec.

12 4 An introduction to ultracold Fermi gases will be discussed extensively in Chapter 3, has many implications in the field of the quantum computing and of the quantum information and will be briefly discussed in Sec Moreover the coupling can be varied also in real time, providing the unprecedented possibility to easily observe a quantum many-body system out of equilibrium. Indeed, at such low temperature, the dynamics is sufficiently slow that a magnetic field sweep across the resonance can be considered almost instantaneous, and the system can be driven out of equilibrium with relative simplicity. Ultracold Fermi gases with tunable coupling provided also an ideal playground to realize a very long-standing theoretical scenario in condensed matter, i.e. the continuous evolution between a weak-coupling BCS (from Bardeen-Cooper-Schrieffer) superfluidity and a strong-coupling BEC (from Bose-Einstein Condensation) superfluidity, i.e. the so-called BCS-BEC crossover. The subject of this thesys will be to investigate up to what extent this crossover can be addresses quantitatively using a lattice model, therefore the next chapter is completely devoted to the discussion of the BCS-BEC crossover in dilute Fermi gases. Later in this chapter instead we will only introduce some theoretical concept for understanding the link between cold atoms and BCS-BEC crossover. The ultracold Fermi gases are nowadays a field with an enormous amount of different lines of research, and it is clearly far beyond the aim of this thesys to present an overview of the field 3. Summarizing, ultracold dilute gases are very appealing from a theoretical point of view, because they provide very clean and tunable systems of remarkable simplicity. Moreover, as we will discussed below and in the next chapter, the physics of dilute systems shows naturally some kind of universality, and it allows to draw conclusion of general validity as in the case of the universal Fermi gas discussed in the last chapters. These results indeed have many implications in completely different fields, ranging from condensed matter to the neutrons stars. Then it is fascinating to think about cold atoms as a very powerful tool to address many fundamental questions about the quantum regime and explore several theoretical scenarios, like e.g. the BCS-BEC crossover, in a quite easy way Some information on ultracold Fermi gas experiments The table (1.1) gives an idea of the typical numbers involved nowadays in a Fermi gas experiments. The gas is prepared and confined using magnetooptical techniques in a multi-step procedure. In every step the temperature is reduced and also the number N of trapped atoms diminishes. In the final stage at T nk, a experiment allows trap up to N 10 7,10 8 atoms. It is important to note that, even though this is likely to be the 3 For review, see [8] and [9].

13 1.1 A (very personal) Overview on Cold Atoms Physics 5 smallest temperature achieved in the universe, they are so dilute that the final temperature is a relevant fraction of the Fermi temperature (T F ρ 2 3, where ρ = N V ). These numbers are only indicative, and specially in the case of optical lattices briefly discussed in Section 1.3, we can have much less particles. The alkaline atoms seems to be suitable for this kind of research, given the intrinsic simplicity which allows to tailor the experimental setup with relative effort. The most used isotopes in Fermi gas experiment are 6 Li and 40 K. These gases are compulsorily not homogeneous systems, because of the unavoidable presence of the trapping potential. From the theoretical point of view, the trapping potential makes quite harder the direct comparison with the experimental data, even if the trap can be taken in account in a second time using the Local Density Approximation (LDA), which we rapidly sketch in Section 2.2. However, in this thesis, we are not dealing with the problem of the trapping potential and all the results presented in Chapters 4 and 5 concern the homogeneous system. On the other side it is well known that the trapping potential can also greatly helps the experimental analysis. In presence of a phase separation, as it happens in the case of polarized Fermi systems, the phase separation manifest itself as a spatial separation. For example, in the first observation of the BEC in 1995, the formation of the condensate phase was clearly evident looking to the height of the density peak at the center of the trap, which showed a striking discontinuous behavior in temperature. As already discussed, nowadays Fano-Feshbach resonances are commonly used to drive the system through different regimes of coupling and this greatly helps also the experimental analysis, because the system can be adiabatically driven to the non-interacting regime where exact results apply. This come to be a very useful tool, for example, for the thermometry of the strongly interacting regime. One of the main difficulties in ultracold gas experiment is to perform measurements to extract of information on the state of the system, once the atoms are cooled. Cold atoms are indeed very fragile and they must be completely isolated from the environment to avoid warming. The majority of spectroscopic methods commonly used in atomic and condensed matter physics would lead to an evaporation of the system. For this reason many information were initially extracted suddenly switching-off the trap and looking for the expansion of the atoms in free space (destructive measure). In recent years instead, new techniques have been developed to probe the sample in a different way and extract more informations. A very promising method seems to be exploiting the interference of two systems prepared in traps close to each other to extract correlations functions [10]. Moreover some specific spectroscopic methods have been developed to address cold atoms. Among them we can cite the Raman [11] and the photoemis-

14 6 An introduction to ultracold Fermi gases sion Radio Frequency spectroscopy [12] 4. These spectroscopic techniques are mainly based on the fact that the atoms are not elementary object and they possess internal degrees of freedom which can be excited using suitable electromagnetic fields. In this sense, the fact that the building blocks of our system are composite objects can be turned to be an advantage. In the next section we will introduce some basic concept for the understanding of the physics of ultracold Fermi gases and dilute Fermi systems in general. 1.2 Basic theoretical tools for cold atoms physics The main property of ultracold gases which greatly simplifies the theoretical approach is that they are dilute, in the sense that the mean interparticle distance is usually much larger than the range of the interparticle potential. The atoms are neutral and, unless we consider atoms with a nonzero electric dipole, leading to a completely different and far less understood physics (see e.g. [13]), at large distances the interparticle potential is of van del Waals form V (r) = C 6 /r 6 for r. (1.1) In principle the power-law behavior (1.1) does not define a typical length scale. However the potential decays fast enough at long distances that some kind of effective range can be operatively defined as R V dw ( mc 6 ), which 2 is the distance at which the kinetic energy of the relative motion of two particles equals their interaction energy. The typical value is of the order of several nano-meters. A complete treatment of atomic collisions is a very complex task and requires a detailed knowledge of the interatomic potential. However, strong simplifications arise in the case of cold dilute atomic gases where all the physics can be expressed in terms of a single universal parameter which is the s-wave scattering length a s that we define below. This simplifications come from quite general properties of the low-energy scattering in quantum mechanics and are expected to be generally valid for dilute systems interacting through some short-range potential. Following [14], if one considers the scattering problem of two particles interacting through a spherical symmetric potential V (r) with range b, the outgoing wavefunction of the relative motion can be written for r b (far field approximation) as ψ + k = e i k r + f(k,θ) eikr r (1.2) where k is the incoming relative momentum and θ is the scattering angle. The scattering state at large distance appears as a superposition of the incoming plane wave and an outgoing spherical wave whose amplitude f(k, θ) 4 The latter method has been recently utilized to extract information about the spectrum of the dilute Fermi gas in the strongly interacting regime.

15 1.2 Basic theoretical tools for cold atoms physics 7 is by definition the scattering amplitude of the process. Using the spherical symmetry the scattering amplitude can be expressed as a sum of various contributes arising from the scattering of different partial waves with angular momentum l = 0,1,2,, i.e. f(k,θ) = (2l + 1)f l (k)p l (cos θ) (1.3) l=0 where P l (cos θ) is the Legendre Polynomial of order l. The amplitude f l (k) in the channel l can be expressed in terms of a phase factor δ l (k) as f l (k) = 1 k cot δ l ik (1.4) To proceed further let us assume a specific shape of the potential which is the spherical barrier (well). V (r) = { V0 for r < b 0 for r b (1.5) with V 0 > 0 (V 0 < 0). If the momentum k is small in the sense that kb << 1 (low-energy processes) is possible to demonstrate that δ l k 2l+1 and the contribution from partial waves relative to high angular momenta are strongly suppressed. The physical explanation is that, if the energy is too low, the incoming wave cannot penetrate the centrifugal barrier whose amplitude is proportional to l(l + 1). This conclusion is expected to be valid in general for short-range potentials even though the specific scaling function depends on the potential at hand. In the limit k 0 only the s-wave contribution stays finite and, for small momentum k, allows for this expansion: lim f(k,θ) = 1 k 0 a 1 s + ik Re 2 k2 +. (1.6) Thus in the limit k 0 the scattering amplitude reduces to a constant value, which is (minus) the scattering length a s, while the first non-universal (ik do not contain any information on the shape of the potential) subleading contribution is negligible if kr e 1. The parameter R e is called effective range of the interaction and do not coincide in general with the original range b. However it can be calculated from the knowledge of the potential solving the scattering problem in the s-wave channel and expanding the result for small k. Intuitively this parameter gives a measure of how small must be k to the scattering processes in the s-wave channel be described only in terms of scattering length a s (k 1 R e ). Even though the specific value of the scattering length depends on the detailed shape of the potential, the last result implies that different shortrange potentials lead to the same low-energy scattering, if they have the same

16 8 An introduction to ultracold Fermi gases scattering length a s. The ultracold gas regime can be operatively defined as the regime where the scattering from higher partial waves can be neglected. This regime, as far as only the scattering length is relevant, naturally shows a kind of universality because all the details of the interparticle potential (but a s ) are irrelevant. Conversely, for a given value of the scattering length (usually taken from ultracold gases experiments), any shape V (r) of the potential is good as another one and this greatly simplify in the theoretical approach, allowing to use simpler models for the interaction, as explained in Sec a_s (b=1) V_0 Figure 1.1: S-wave scattering length (b = 1) for the square well (barrier) potential of (1.5) as a function of V 0 ( = m = 1). Still considering, for simplicity, the square barrier (well) of range b defined in (1.5), the scattering length is given by [ ] b 1 tanh(k 0b) (k a s = 0 b) for V 0 > 0 [ ] b 1 tan(k (1.7) 0b) (k 0 b) for V 0 < 0 where k 2 0 = mv 0 2. With reference to Fig. 1.2, on the repulsive side (V 0 > 0) the scattering length is positive and increases monotonically with V 0. For the hard-sphere potential the scattering length coincides with the range of the potential itself (lim V0 a s (V 0 ) = b). On the attractive side (V 0 < 0) the value of the scattering length is very sensitive to the presence of bound states close to zero energy and can be also much greater than the range of the potential and also change sign. If we increase V 0 starting from zero

17 1.2 Basic theoretical tools for cold atoms physics 9 the scattering length is first negative and small. When V 0 crosses the minimum value for which a bound state can be formed (which is finite in 3d) the scattering length is negatively divergent. Crossing the critical coupling for the bound state the scattering length change sign and diverges positively and then start decrease until V 0 approaches the next bound state where it diverges again, and so on. This connecting between the divergences of the scattering length and zero energy bound states is completely general. Therefore a very detailed knowledge of the real potential is required to calculate from first principle the actual value of the scattering length because it depends critically on the location of bound states. Thus, as already pointed out, it is much easier to assume the scattering length as given by experiments and employ some kind of model for the interaction leading to the same scattering length. So far we did not use at all the quantum statistics and the previous considerations apply both to Fermions and Bosons. If we consider Fermionic atoms in the same hyperfine state, then the antisymmetry constraint imposes that the scattering contribution coming from even partial waves l = 0,2,4, must be zero. In this case the leading contribution is the p- wave which is suppressed at low-energy, leading to non-interacting polarized Fermions in the dilute ultracold regime, as discussed in the previous section. Thus, to have a finite leading contribution, it is necessary to have two hyperfine states interacting with each other. The intraspecies scattering will be suppressed for the same reason as above, and the physics will depends only on the interspecies scattering length. Strictly speaking, all these considerations apply only to the two-body problem at low energy. The conceptual link with the ground state of a many-body system of dilute particles is somewhat tricky. In the case of a purely repulsive potential, it is well-known [15] that for a dilute Fermi system, i.e. a gas of short-range interacting fermions with k F b 1, where k F = (3π 2 n) 1 3 is the Fermi momentum of the system and b is the bare range of the interparticle potential, the ground state energy can be expanded in powers of the so-called gas parameter k F a s, i.e. E N = 3 5 E F ( (11 2log 2) 9π 2(k Fa s ) + 21π 2 (k F a s ) 2 ). (1.8) Thus the scattering length a s and the Fermi momentum k F naturally appear in the problem via the gas parameter. It is important to point out that being a s < b for a purely repulsive potential, the dilution condition imply also that the gas parameter is small and the perturbative series is fast convergent. On the attractive side the gas parameter k F a s is still expected to be the relevant variable for describing the physics of a dilute gas. However, the condition k F a s 1 is not anymore satisfied if the scattering length is large, as usually happens close to a zero-energy bound state. In this condition much more complicated theories have to be used, as we will see in the next chapters.

18 10 An introduction to ultracold Fermi gases Microscopic models for the interaction The relevance of only the scattering length a s in the description of the dilute systems leads to the conclusion that the model potential can be chosen as the simplest as possible giving the right value of a s. To eliminate the range b of the potential from the problem, which would make the dilution condition k F b 1 fulfilled for every density, the easiest choice would be take a zero range delta potential V (r) = gδ(r). However, according to the above arguments, one is lead to conclude that in 3d this kind of potential does not give any scattering process, i.e. it is transparent. This can be easily understood considering the limit b 0 of a hard-sphere potential. This kind of potential would introduce unphysical ultraviolet (UV) divergences in perturbative expansions like (1.8), requiring some kind of regularization. Several choices have been proposed to regularize the interaction. The most used is the zero-range pseudopotential [16], where pseudo means that the mathematical form of this potential include some kind of regularization condition to overcome the problem discussed above. This condition is r V (r) ψ( r) = gδ(r) [ r (rψ(r)) ] r=0 where g = 4π 2 a s m (1.9) This kind of pseudopotential was first introduced by Fermi in his study of slow neutrons. The remarkable simplicity of having a coupling constant g proportional to the scattering length a s is partially spoiled by the fact that the potential admits only one one bound state when the interaction is repulsive, which is partially counterintuitive and clearly stems from the regularization condition. Another, maybe more transparent, way to regularize the problem is to define a cut-off k 0 leading to the right scattering properties. In momentum space the Fourier transform of the scattering potential can be taken to be { g for q < k0 V ( q) = (1.10) 0 for q k 0 The scattering length a s defined in (1.6) can be also obtained as the limit for small momentum and energy of the scattering T-matrix, which is the sum of ladder diagrams [15], calculated for µ = 0 (i.e. zero chemical potential which corresponds to the two-particles problem). For the potential (1.10) the expression is particularly simple and gives m 4πa s = g 1 + k k 0 d k (2π) 3 m k 2 for = 1, (1.11) which relates g to k 0 and a s 5. If we remove the UV cut-off (k 0 ) for fixed g 0 the integral diverges, leading to the condition that the scattering 5 This formulation will be used in Chapter 4 to relate the coupling U to the scattering length a s in our lattice model.

19 1.2 Basic theoretical tools for cold atoms physics 11 length is zero for every finite value of the interaction. However it is possible to remove the cut-off at fixed scattering length a s, if g 0 while k 0. Even though this kinds of model potential are extremely simple, they are not very easy to implement in a numerical calculation, like Quantum Monte Carlo simulations for example. In that case, people usually chose some kind of short-range potential of range b, like e.g. the square well of the previous section. However, explicitly introducing the range b of the interparticle potential, the dilute limit (k F b 1) relevant for ultracold atoms is recovered only for low enough densities Fano-Feshbach Resonance Figure 1.2: Schematic picture of the Fano-Feshbach resonance. The Fano-Feshbach resonance is well-known from longtime in the context of atomic and nuclear physics. Indeed, first U. Fano in 1935 [17] and then H. Feshbach in 1962 [18] studied this phenomenon in the context of atomic physics and of the nuclear physics respectively. In this section we try to give a very sketchy picture of the phenomenon. For cold atoms this phenomenon arise in presence of two different hyperfine states with a different magnetic momentum. With reference to Fig , the scattering channels associated to different total hyperfine state, called the open channel (bottom line) and the closed channel (upper line), are coupled because during a collision the atoms can be promoted from one channel to the other. Intuitively this coupling depends of the distance in energy between the continuum threshold

20 12 An introduction to ultracold Fermi gases of the scattering channel (open channel) and the bound states of the closed channel. Because the molecules in different channels have a different magnetic momentum, in presence of a magnetic field their energy will be split in a different way and it is possible to drive this systems to a condition where these energy levels coincide. In this condition the two channels are resonantly coupled and further increasing the magnetic field the energy levels cross each other and the initial situation is reversed. A complete description of the resonance is far beyond the scope of this brief overview and requires in principle a multichannel treatment. However, for the connection with the treatment of the crossover BCS-BEC in condensed matter, a key point is that, in presence of a broad Fano-Feshbach resonance [19] (R e k >> 1, where R e is the effective range of the interaction and k k F for weak to intermediate coupling and k a 1 s for strong coupling), the problem can be effectively described within a single channel model with a scattering length depending on the external magnetic field B, i.e. ) w a s (B) = a bg (1 B B 0, (1.12) where a bg is the background (non-resonant) scattering length for atoms scattering in the open channel, B 0 is the value of magnetic field at which the molecular bound state of the coupled channel crosses the continuum threshold of the open channel, and w is the width of the resonance, defined as the distance in magnetic field between B 0 and the magnetic field at which a s = 0. Summarizing in presence of a broad Fano-Feshbach resonance (e.g. like the one occurring in the case of 6 Li at 822G), the system can be described as a system of Fermions with two hyperfine states (let s say spins) interacting through a potential with a scattering length given by (1.12). Thus the BCS-BEC crossover for Fermi atoms with a broad Fano-Feshbach resonance is placed on the same footing as the corresponding crossover in different physical systems, once the scattering length is assumed as the relevant variable for describing the interaction in dilute systems. As pointed out in the previous section, the divergence of the scattering length is generically associated to the presence of a bound (or virtual) state near the continuum threshold, so, in principle, an attractive potential with tunable coupling is enough to describe this kind of resonance. It is quite intuitive that the presence of a resonant scattering phenomena will affect the rate of scattering processes and eventually the lifetime of the system. Yet, as already pointed out in the first section, we are very lucky for the case of Fermions 6, as opposed to the case of Bosons [20], and near the resonance Fermions shows a remarkable collisional stability which allows an easy experimental observation of the metastable gas phase. 6 The theoretical explanation of this phenomenon was given by D.Petrov in [6]

21 1.3 Optical lattices and idealized solid state hamiltonians Optical lattices and idealized solid state hamiltonians In recent years it becomes possible to realize controllable and tunable systems that, under suitable conditions, are expected to be almost ideal realizations of popular models of correlated electrons on a lattice, like the Hubbard model [21] (attractive or repulsive). As already mentioned, the Fano-Feshbach resonance can be used to tune the interaction strength. The presence of the ionic lattice is a key feature in the physics of condensed matter. An optical version of the periodic potential can be realized using two counterpropagating laser beams, since, for far-detuned laser beams, the main effect of the laser field on the atoms is to induce an electric dipole. The standing wave created by the lasers determines a periodic potential for the atoms, with a strength proportional to the intensity I of the beam. The lattice spacing is instead controlled by the wavelength of the lasers. In this way a system of Fermions in different hyperfine states (to exploit the Fano-Feshbach resonance), can be effectively used to mimic the behavior of the electrons with different spins in solids. This tool is really very versatile, and it allows to realize with amazing simplicity systems of Fermions loaded in lattice of different types (cubic, triangular, etc.) and also to vary the dimension of the system exploiting strong anisotropies in the periodic potential [22]. Moreover, compared with the ionic lattice, there are no dislocations and phonons. We can therefore realize a quantum system that resembles closely simple models, and that with some optimism can be used to simulate the properties of these models in a lab. This has an obvious relevance from a theoretical point of view, because provides an unbiased test of the approximated techniques that people developed to treat this model and strongly correlated electronic systems in general. It is remarkable that ultracold atoms in a suitable setup can be described by the Hubbard model much better than electrons in strongly correlated solids. Many paradigmatic situations in condensed matter have already been realized in optical lattices, like the progressive building of the Fermi surface in a lattice, which eventually leads to a band insulator [23] and the observation of Bloch oscillations induced by gravity [24]. Recently people finally worked out in the realization of the Hubbard model and also of the Mott-transition, i.e. the metal-insulator transition driven by the interaction [25]. However, there are still a lot of open questions and many technical issues remain unsolved. To reach interesting regime for condensed matter, as pointed out in [26], far lower temperature are needed. Moreover the same thermometry for cold atoms in a lattice is an open question. In this thesys we will mainly present results for the attractive Hubbard model. Yet, we will not directly address the case of optical lattices, and the this model will be only used to address the problem of the BCS-BEC

22 14 An introduction to ultracold Fermi gases crossover in homogeneous space discussed in Chapter 2, troughout a suitable extrapolation of the dilute limit as explained in Chapter 4. A brief conclusion Ultracold Fermi atoms are a very useful tool to address many-body systems. Among the big variety of lines of research, in this thesis we will focus only on the BCS-BEC crossover. As already discussed, mixtures of ultracold fermionic gases in different hyperfine states interacting through a (large) Fano-Feshbach resonance provide an extremely simple realization of the crossover. The physics of the system can be described in terms of the only gas parameter k F a s, where the scattering length a s can be tuned by applying a magnetic field using the resonance. In this way we obtain a dilute Fermi system with tunable interaction. The next chapter will be entirely devoted to the crossover BCS-BEC with a special focus on the crossover in dilute systems.

23 Chapter 2 The BCS-BEC Crossover 2.1 Brief History of a Long-Standing Problem Quantum mechanics teaches us that all the particles are either Fermions or Bosons and their statistical behavior in a system composed of many identical particles is dictated by their fermionic or bosonic nature. This is clearly the case for elementary particles, like electrons or photons. If instead the basic blocks as well are composite object (e.g. bosonic atoms formed by electrons, protons and neutrons or electron pairs) their statistical behavior can change according to the overlap between the constituents. If these objects are far apart then they will see each other as point-like particles and the statistical properties will be fixed only by the total number of the constituent fermions, i.e. they can be safely considered composite bosons. However, if they come very close to each other the antisymmetric statistics of constituent fermions will eventually come to play a role. The history of fermionic superfluidity (or superconductivity in the case of charged particles) could also be reanalyzed from this point of view. Superconductivity was discovered in 1911 by H.K. Onnes [27], who cooled a metallic sample of mercury finding no electrical resistivity below 4.2 K. Experimental evidence for superfluid behavior in bosonic 4 He emerged only in 1938 when two independent measurement of viscosity were reported, one by P.Kapitsa [28] and the other by J.Allen and D.Misener [29]; the measurement indicated exceedingly low, essentially vanishing viscosity below 2.2 K. Not long after Fritz London suggested the connections between superfluidity and Bose-Einstein condensation [30]. Some microscopic understanding was developed by N.N. Bogolubov, who demonstrated that the excitation spectrum of a weakly interacting Bose gas [31] grows linearly in the momentum of the excitation and the critical velocity is finite. Then a weakly repulsive interaction do not destroy the Bose-Einstein condensate, and this interaction is fundamental to have a superfluid because to have a nonzero critical

24 16 The BCS-BEC Crossover velocity correlation are required. Drawing from the lessons of superfluid Helium and the connection to BEC, Max Schafroth proposed in 1954 that superconductivity in metals was due to the existence of a charged Bose gas of two-electron bound states that condense below a critical temperature [32]. However, experiment did not seem to support that simple picture. It was not until 1956 that a key idea emerged: Leon Cooper discovered [33] that an arbitrarily small attractive interaction between two electrons of opposite spins and opposite momenta could lead to the formation of bound pairs in presence of many other Fermions, in striking contrast to what happens for two-body problem in three-dimensional free space, where only beyond some coupling threshold particles can form a bound state. The presence of the other Fermions, which actually limit via the Pauli principle the available phase-space, plays a key role. Since pairing occurs in momentum space and the attraction between the electrons is weak, the typical size of the pairs is quite large compared to the mean interparticle distance and they cannot be regarded as individual bosons like in the Schafroth s picture. The world renowned microscopic theory of conventional superconducturs (BCS theory from J.Bardeen, L.Cooper and R. Schrieffer [34]) dates back to Their picture incorporates in a true many-body state the Cooper idea of large overlapping fermions pairs. The BCS theory was an immediate success, since it could explain many experimental results of the time at a good quantitative level. As already emphasized by the authors, their theory did not describe the picture proposed by Schafroth, Butler and J.Blatt [35], also in 1957, in which Bose molecules, local pairs of electrons with opposite spins, form an interacting charged Bose gas that condenses and becomes a superconductor. Thus in those days it was the differences - rather than the similarities - between these alternative points of view which one emphasized, so that they seemed incompatible with each other. It is necessary to wait until 1969 when Eagles [36] first posed the question of the strong-coupling limit of the BCS theory in the context of superconducting semiconductors. He realized that the strong coupling limit of the BCS gap equation was identical to the condition for having a bound state of Fermions of binding energy E = µ, where the chemical potential µ in this limit is strongly affected by the interaction and the approximation µ E F is not anymore satisfied. Amazingly, a clear picture of the evolution from weak to strong coupling aimed to reconcile the picture of BCS Cooper pairs and Schafroth BEC emerged only in 1980, when Anthony Leggett in his paper Diatomic molecules and Cooper pairs [37] realized that a unifying point of view could be captured by a simple description in real space of paired fermions with opposite spins 1. Leggett considered a dilute gas of fermions in a mean-field framework and shown that the evolution of the ground state, 1 The Leggett theory is briefly summarized below because of the relevance for the results presented in Chapters 4 and 5.