Exploring spin-orbital models with cold gases loaded in p-bands of zig-zag optical lattice

Transcription

1 Exploring spin-orbital models with cold gases loaded in p-bands of zig-zag optical lattice Temo Vekua Institut für Theoretische Physik Leibniz Universität Hannover in collaboration with: G. Sun, G. Jackeli and L. Santos Goslar, 13 February 2013 p. 1

2 Spin-orbit couplings in solids Relativistic spin-orbit coupling (e.g. compounds with strong Rashba or Dresselhaus coupling): Dirac equation for electron in external potentialv expanded to lowest order inv 2 /c 2 leads to a Schrödinger equation with additional spin-orbit coupling term: [ V p] S 2m 2 ec 2 1 2m 2 ec 2 r V r ( S L) Non-relativistic spin-orbit coupling (Kugel-Khomskii): If relevant electrons are from p, d, or f-bands and angular momentum is not quenched there is orbital degeneracy. Spin-orbit coupling can give nontrivial topology to band insulators, with helical gapless edge modes or can induce topological superconductivity with Majorana particles. Goslar, 13 February 2013 p. 2

3 Solid with zig-zag structure In Mott insulators orbital degrees enhance thermal as well as quantum fluctuations, suppress Néel order, and lead to spin/orbital liquid states. Mott insulating transition-metal compounds having structures of weakly coupled zig-zag chains: with strong relativistic coupling : Sr 2 VO 4, the role of spins is played by an isospin variable discerning the Kramers partners, while the pseudo-orbital variables distinguishes two lowest Kramers doublets of V 4+ ion. with negligible relativistic coupling: Pyroxene titanium oxidesatisi 2 O 6 (A = Na,Li) with actived xy andd yz orbitals, and spin one-half from Ti 3+ ions. Goslar, 13 February 2013 p. 3

4 Advantages of ultra-cold gases In many solid-state systems, the orbital dynamics is often quenched, due to the coupling of the orbitals to Jahn-Teller phonons, and the quantum nature of orbitals is lost. In real materials, the coupling strengths are fixed by nature, being very difficult to modify (e.g. by pressure), thus the experimental access to a potentially rich phase diagram is limited. Ultra-cold dipolar spinor fermions in zig-zag type optical lattices can mimic spin-orbital models relevant in solid-state systems with the interesting advantage of reviving the quantum nature of orbital fluctuations and exploring vast phase diagram. Goslar, 13 February 2013 p. 4

5 Zig-zag chains of optical lattice i+1 i i+2 Incoherent superposition between a triangular lattice V 1 ( r (x,y)) ( = ) V 10 [sin 2 b1 r/2 ( ( +sin 2 b2 r/2 )+sin 2 ( b 1 )] b 2 ) r/2, withk the laser wavenumber b1 = 3k e y and b 2 = 3k( 3 e x /2 e y /2) and an additional lattice V 2 ( r) = V 20 sin 2 ( 3ky/4 π/4). On FigureV 20 /V 10 = 2. Goslar, 13 February 2013 p. 5

6 P-bands of optical lattice P x P y S P z band is energetically higher due to tight confinement in z direction. σ x = +1 P x orbital, σ x = 1 P y orbital. Goslar, 13 February 2013 p. 6

7 P-bands of optical lattice P x P y S P z band is energetically higher due to tight confinement in z direction. σ x = +1 P x orbital, σ x = 1 P y orbital. Py 0 P x U λ 1 t 2 V 3 U, V are on-site interaction energies,t(>0) andλ are hoppings. λ 0 e.g. due to elongating confining potential in x-y plane along e x + e y direction. Goslar, 13 February 2013 p. 6

8 Dipolar Fermions Magnetic dipoles, like Chromium atoms from ground state F=9/2 manifold in strong magnetic field. Polar fermionic molecules with strong electric dipole moment e.g., 40 K 87 Rb or 7 Li 40 K. Spins and are provided by two internal states of fermion that are distinguished by nuclear spin, which in the lowest energy states decouples from electrons, and hence the interactions (which are due to electronic degrees of freedom) are spin independent,su(2): V(r 1 r 2 )n(r 1 )n(r 2 ), n = n +n. Total interparticle potential inlcudes dipolar and contact terms: V (r 1 r 2 ) = V α,β (r 1 r 2 ) = µ 0d α d β 4π r 1 r 2 3 +gδ(r 1 r 2 ) For magnetic dipoles, in weak fields, the interparticle potential will depend on the hyperfine state, as different hyperfine components have different magnetic momentsd d. Goslar, 13 February 2013 p. 7

9 53 Cr Breit-Rabi diagram Fermionic chromium 53 Cr: I=3/2, J=S=3 (Hund), F=9/2,...3/2. For strong magnetic fields, the magnetic dipole moment of 4 lowest energy states 9 2, 9 2, 9 2, 7, 2 9 2, 5, 2 and 9 2, 3, are 2 equald α d β 2µ B m J / = 6µ B. E 3/2 5/2 7/2 9/2 h z m =S = 3 J } I=3/2 d 2 = 36µ 2 B for 53 Cr, compare tod 2 = µ 2 B for alkali metals with J = S = 1/2 in the ground state. Erbiumd 2 = 49µ 2 B and dysprosiumd 2 = 100µ 2 B. Goslar, 13 February 2013 p. 8

10 Dipolar Fermions The interaction energy of two fermions occupying the same orbital of a given lattice site (necessarily in spin singlet state) is U, U= dr 1 dr 2 P 2 x(y)(r 1 )V(r 1 r 2 )P 2 x(y)(r 2 ), when two fermions occupy different orbitals of the same lattice site they can be either in spin triplet or spin singlet state. Define average interaction energyv, V = dr 1 dr 2 P 2 x(r 1 )V(r 1 r 2 )P 2 y(r 2 ), P x andp y are orbital wavefunctions at the same well. Goslar, 13 February 2013 p. 9

11 Dipolar fermions in P-bands Interaction in spin-triplet state is V t = V J H Interaction in spin-singlet state is,v s = V +J H. J H = dr 1 dr 2 P x (r 1 )P y (r 1 )V(r 1 r 2 )P x (r 2 )P y (r 2 ) For the spin-triplet state, the contacts-wave interaction does not play any role, V(r 1 r 2 ) δ(r 1 r 2 ), V = J H and scattering in the spin-triplet channel vanishesv t = 0. In general for Coulomb, dipolar, contact, other potential one can showv t < V s Hunds rule: by maximazing spin of two fermions on orthogonal orbitals, the energy (of Coulomb, dipolar or contact repulsion) is reduced, since fermions minimize their overlap in coordinate antisymmetric state (that has a node at vanishing relative distance). Goslar, 13 February 2013 p. 10

12 Dipolar fermions in P-bands Goslar, 13 February 2013 p. 11

13 Dipolar fermions in P-bands In strong coupling U,V ±J H t,λ and for one fermion per lattice site spin-orbit model. Goslar, 13 February 2013 p. 11

14 Dipolar fermions in P-bands In strong coupling U,V ±J H t,λ and for one fermion per lattice site spin-orbit model. H = t2 2U (2S i S i+1 +α 1 2 )(1+( 1)i σi)(1+( 1) x i σi+1) x i i 2S i S i+1 (1 σ x iσ x i+1) λ i σ z i α = U(V+J H/2) U/V and = J V 2 JH 2 H U/(V 2 JH 2). Possibility ofλ 0 gives to orbitals quantum character. Note, despite that we restrict interactions within the same lattice site, dipolar (long-range) interaction is crucial for the stability of the Mott 1 state. For contact potentials one would get metalic ferromagnetizm, since. Observe SU(2) symmetry in spins, due tov(r 1 r 2 ) being independent of spin projection. Goslar, 13 February 2013 p. 11

15 Ground state phases forα < 2 Goslar, 13 February 2013 p. 12

16 Ground state phases forα < 2 λ (D,F) (ih,p) (a) (ih,af) α (a) = 0, (b)α = 1 forα = U(V+J H/2) < 2 the similar topology of phase diagram. V 2 JH 2 Ground state is spin singlet,s T = 0 or fully polarizeds T = 1/2 in Ferro phase. λ (ih,p) (D,F) (D,F) (b) (F,P) (F,AF) Goslar, 13 February 2013 p. 12

17 Ground state phases forα > 2 Goslar, 13 February 2013 p. 13

18 Ground state phases forα > 2 λ ih,para ih,af α>2 F,AF Ferrimagnet T 0<S <1/2 F,Para for α > 2 the similar topology of phase diagram. Ground state is global spin singlet, in ih states and fully polarized in Ferro phases In Ferrimagnetic phase total spin of the ground state changes smoothly and takes all values from S T = 0 tills T = 1/2. Interplay betweenλ(singlet formation) and (Hund triplet formation) produces Ferrimagnetic state. Goslar, 13 February 2013 p. 13

19 2 fermions per site The case of 2 fermions per site does not need dipolar interactions. Due to Hunds rule on-site triplet statess T = 1 will be favoured (interaction energy in spin-triplet state vanishes for local interactions) and low energy effective model is Haldane S = 1 spin chain: H = J i S i S i+1. Ground state is that of AKLT type. Goslar, 13 February 2013 p. 14

20 Spin exchange scale Goslar, 13 February 2013 p. 15

21 Spin exchange scale 2-component Hubbard model: t U ForU t, retain one particle per site. Second order perturbation theory in tunneling, H Heisenberg = ±JS 1 S 2, S i = 1 2 b i,α σ α,βb i,β + sign for fermions antiferromagnetism, opposite to Hund rule, gain is due to delocalization energy - sign for bosons ferromagnetism. J = 4t2 U strength of spin exchange. In cold gases currently U > T > J. Goslar, 13 February 2013 p. 15

22 Resolving spin degeneracy ForU T J, probability that spin looks up or down is 50%, independent of its neighbors,s = k B ln2. To resolve spin coherence for 3D antiferromagnets on cubic lattice,t < T N J, s(t N ) 0.5ln2k B (QMC). To observe AFM spin orders < 0.5ln2k B. Current experimentss ln2k B (spin incoherent Mott state). How to get so small entropy per site? Start from Band insulator state with practically zero entropy (boundary defects and uncertainty of double occupancy will give some residual but small entropy). Use spin-changing collisions and orbital degeneracy. Goslar, 13 February 2013 p. 16

23 Towards the spin degeneracy Use internal degrees of freedom (hyperfine spin components) and spin-changing collisions. 40 K has 10 states in the lowest hyperfine multiplet. 9/2 7/2 1/2 3/2 Initially F = 9 2,m F = 9 2 and 9 2, 1 2 both in s-orbital finaly 9 2, 7 2 and 9 2, 3 2 in p-orbital. In experiments currentlyλ t. Goslar, 13 February 2013 p. 17

24 Topological Haldane state Take multi-component fermions and initially select 2 components by magnetic field in band insulator state. Quench magnetic field to resonant value (to allow production of other two components) and after one cycle, t = t, quench it again away of the resonant value. Goslar, 13 February 2013 p. 18

25 Topological Haldane state Take multi-component fermions and initially select 2 components by magnetic field in band insulator state. Quench magnetic field to resonant value (to allow production of other two components) and after one cycle, t = t, quench it again away of the resonant value. 1.0 populations /2, 1/2 time after quench /2,7/2 t* One can load with high efficiency p-bands and allow to time evolve the system. Since effective Hamiltonian is of Haldane spin-1 chain one can arrive at Haldane state. Adiabatic change of field talk by Juan Jaramillo. Goslar, 13 February 2013 p. 18

26 Vielen Dank! Goslar, 13 February 2013 p. 19