The fermionic Hubbard model in an optical lattice

Transcription

1 The fermonc Hubbard model n an optcal lattce Pnak Majumdar Harsh-Chandra Research Insttute (HRI) Allahabad, Inda QIPA School: Feb 2011

2 Plan of the lecture Background: many body problems Repulsve Hubbard model -A. Conjectured phase dagram -B. The Neel state and Mott physcs -C. Effect of geometrc frustraton -D. Mott shells n trapped fermons -E. d-wave superfludty?! Attractve Hubbard model -A. Overall phase dagram -B. BCS-BEC crossover n superfluds -C. Superflud- nsulator transtons -D. Inhomogeneous superflud n a trap Concluson

3 I. Many body systems

4 1. Weakly nteractng many body systems Smplest case: free ferm and bose systems! - many partcle states are product states -many body character enters only through exchange statstcs -fermons: C V N(ɛ F )T, χ N(ɛ F ), no phase transtons.. -bosons: macroscopc occupaton of ɛ 0 at T h2 2m n2/3.. BEC. The mpact of weak nteracton: () normal state. -the normal state corresponds to absence of any long range order.. -can use perturbaton theory for ground state energy, dampng.. -hghly developed theory, but needs a small parameter! The mpact of weak nteracton: () phase transtons, order.. -even weak nteractons can lead to orderng -weak attracton parng and superfludty -weak repulson can sometmes lead to antferromagnetsm -requres self-consstent treatment: mean feld (MF) theory.. Many systems cannot be understood wthn perturbaton/mf theory..

5 2. Correlated systems Present effort: understand phenomena beyond straghtforward perturbaton theory. The physcs cannot be vsualsed n terms of ndependent degrees of freedom. Examples? Superflud to Mott nsulator transton n bose systems. Mott state and magnetc order for repulsve fermons. Possble superfludty n doped fermonc Mott systems. The BEC state n attractve fermonc systems. Mott state: jammng! Moton of doped holes.. superposton..

6 3. Models Consder two artfcally smple models of correlaton. These are approxmately realsed n the sold state. They can be engneered n optcal lattces! The repulsve (+ve U) Hubbard model H = t j c σ c jσ + j σ (V µ)c σ c σ + U n n The attractve (-ve U) Hubbard model H = t j c σ c jσ + j σ (V µ)c σ c σ U n n t j denote hoppng on the lattce, V s a potental.. ref fg.. If U = 0 we can dagonalse H n terms of Bloch states.. For t j = 0, the system s dagonal n a purely local bass.. If both t j and U are non-zero.. no exact soluton for d > 1.

7 4. How s the model solved? Mean feld theory: -factorse Un n U c c c c or U c c c c -superconductng or magnetc correlatons.. compute self-consstently. -easy to mplement, possbly large fluctuatons.. ncorrect n low d. Problems nvolvng coupled quantum varables admt only two exact methods: -() Exact dagonalsaton (ED), () Quantum Monte Carlo (QMC) ED s severely sze lmted. For a model wth m bass states per ste, and N stes, the matrx sze m N e Nlnm. Wth great effort one can solve for N 16 for m = 4 (Hubbard model), explotng all possble symmetres. Fermon QMC s usually mplemented by rewrtng the nteracton n terms of an auxlary HS varable, φ(r, τ), say. - non-nteractng fermon problem s solved for sampled confgs of φ(r, τ). -convergence s poor at low temperature due to the fermon sgn problem. -state of art 8 8 on Hubbard, low T s naccessble. ED and QMC are also used as solvers wthn dynamcal mean feld theory (DMFT).

8 5. Hubbard physcs: an outlne Why Hubbard? Smplest model of quantum correlaton. Sngle fermonc band wth local nteracton. The attractve model descrbes s-wave superconductvty. The repulsve model descrbes magnets and Mott nsulators. We wll explore the followng ssues n the context of ths model: () strong couplng, () randomness, () confnement, and (v) frustraton Optcal realsaton: () lattce and trap, () bosons/fermons, () tunable nteracton! Expermental status.. -SF to Mott nsulator transton n bosons -Strongly nteractng ferm gases -s-wave superfludty of fermons Challenges: () AF order n fermonc Mott systems, () d-wave superfludty..

9 II. Repulsve Hubbard model

10 A. Conjectured phase dagram The repulsve (+ve U) Hubbard model: H = j,σ t j c σ c jσ + (V µ)n + U n n Serves as the mnmal model for Mott transton and antferromagnetsm. Also extensvely studed as a canddate for d-wave superconductvty. The model s charactersed by: NN hoppng t, and longer range hoppngs, t, t, etc nteracton U, we wll use U/t as measure of couplng the densty, n, controlled by µ possble dsorder or confnng potental V.

11 Phases? at n = 1 and NN hoppng, AF nsulator (Slater to Mott crossover) at n = 1 on a frustrated lattce: complex magnetc order/mi trans for n 1 metal/superconductor/magnet..? no convncng theory n a confnng potental: Mott shells, metal-nsulator coexstence Fgure 1: Dopng-temperature phase dag at large U/t.. cuprates.. How do we proceed? many approx methods, we use the followng...

12 Recast as quadratc fermon problem! Rewrte n n n terms of n 2 and/or σ 2 z. So? Stll quartc? n n = 1 2 (n n ) (n + n ) n n = (n + n ) (n + n ) n n = 1 4 (n + n ) (n n ) 2 We can use the dentty below to lnearse an operator.. Hubbard-Stratonovch (HS) [ ] 1 exp 2 A2 = [ ] 2π dy exp y2 2 ya In Z, ntroduce a new varable at each (space-tme) pont, to be traced over. For example, use φ couplng to n and m couplng to σ z..

13 We use the followng (approxmate) representaton: H = t j c σ c jσ + (V µ)n + U n n = H 0 + U n n j,σ Magnetc decomposton of the model.. H H 0 + [φ n 2 m. σ ] + φ 2 U + m 2 U The φ s a problem! replace by saddle pont value.. left wth vector aux feld m.

14 B. The Neel state and Mott physcs Consder the model on a square lattce wth nearest neghbour hoppng t. H = t c σ c jσ + U j,σ n n At half-fllng the ground state s nsulatng wth {π, π} AF order.. motvate.. Fnal effectve Hamltonan at half fllng: fermons + classcal HS feld m. H = H kn U 2 m σ + U 4 m 2

15 Physcs of the AF crossover: MFT and fluctuaton effects.. -Bpartte lattce: nestng drven small U SDW state, Slater nsulator. -Large U, localsed electrons, Mott state, superexch drven Hes AF. -Wth ncreasng T the weak U system loses AF order and ns character. -Large U, low T c Neel state, paramagnetc nsulator above T c. Evoluton of the DOS wth U and temperature.

16 Fermons coupled to local moments m. The sze of the moments depend on U, and also on T. Local moment formaton at a U dep temperature scale.. Comment on the weak to strong couplng crossover..

17 C. Effect of geometrc frustraton H = t j,σ c σ c jσ t j,σ c σ c jσ + U n n Left fg shows an trangular lattce, rght: equv square lattce. We consder the square wth NN hoppng t and dag hoppng t. Ths s the ansotropc trangular lattce, t = t sotropc lattce. We have seen that for t = 0, n = 1 s an AF nsulator. t frustrates AF order, consequence?

18 U G-AF SP TRI U U m = t t Left: phase dagram. Order destroyed by t at small U. Mddle: sze of the local moment, red: small, yellow: large. Rght: varaton of the T c, lght -large, dark -small. Reproduces known results, captures new fnte T features. Metal-nsulator transtons n the ground state, contrast t = 0... Unusual transport (Hall response) due to non-coplanar spn confgs..

19 D. Mott shells n trapped fermons Test case for nhomogenety: Hubbard model + harmonc potental. We explore the effect of a trap: V = V 0 (x 2 + y 2 ) on the Hubbard lattce. Why nterestng? -The magnetc order n the Hubbard model s robust only at n = 1. -n = 1 mples n = 1 n a homogeneous (flat) system. -In a trap, n would be larger at the center and small at the boundary! -How wll the magnetc order/mott character show up n such systems?! Method: treat φ n terms of a thermal average..

20 Varaton of the densty n r (left), moment m r (center) and NN m r. m r (rght). Top row: densty plateau s at n = 1 and n = 0, correspondng m r and m r. m r. Bottom: plateau s at n = 2, n = 1 and n = 0, and correspondng m r and m r. m r.

21 Results based on R-DMFT Gorelk et al., PRL (2010).

22 E. d-wave superfludty of fermons AF and superconductng correlatons... Chesa et al., PRL (2011)

23 III. Attractve Hubbard model

24 A. Overall phase dagram The -ve U Hubbard model s the smplest example of attractve nteracton. Somewhat artfcal n condensed matter, more real n cold atoms. H = t j σ (c σ c jσ + h.c.) U n n µ σ n σ Involves electrons/atoms hoppng between lattce stes and an on-ste attracton. At all denstes except n = 1 the ground state s superflud. At n = 1 charge densty wave (CDW) and superflud correlatons coexst. Weak couplng, U/zt 1: momentum space parng, BCS superflud Strong couplng, U/zt 1: BEC of molecular pars (more soon) Optcal lattces nowadays allow tunng of the nteracton strength. They also brng n new features: nhomogenety, spn mbalance..

25 B. BCS-BEC crossover n superfluds The method, quckly: H = t j σ (c σ c jσ + h.c.) U n n µ σ n σ HS decouple n the Cooper channel (why!), neglect dynamcs.. H H kn + ( c c + c c ) + 2 U = φ e θ s a complex scalar feld. H s now quadratc n fermons for any confguraton { }. Can be solved va a Bogolyubov transformaton (next slde) BCS works exactly the same way, assumes =, constant.

26 Transform c = (u nγ n v n γ n ), c = (u nγ n + v n γ n ) Canoncal tranformaton mples n ( u n 2 + v n 2 ) = 1 and H = E n + σ E n γ nσγ nσ E n ( ) are nonnegatve egenvalues of the BdG equatons Effectve Hamltonan n terms of the HS felds H eff { } = 2 U E n 1 β n n ln(1 exp( βe n )) The expanson of ths, for unform, leads to Landau theory. How do we know the equlbrum confguratons of {φ, θ }? Cluster MC.. Mean feld/bcs: phase locked unform state, superflud as long as 0. Statc HS MFT for T 0, but very dfferent at strong couplng and T 0.

27 Let us characterse the superflud state n terms of (1) (0), the gap at T = 0. (2) ξ(0), the T = 0 coherence length. (3) the transton temperature: T c Tc U Fgure 2: Left- T dep of SF order parameter. Rght- non-monotonc T c (U). U/zt 1: BCS parng, (0) e 1/N(ɛ F )U, ξ(0)/a 0 1, k B T c / (0) 3.5 U/zt 1: (0) ncreases, smaller ξ(0), k B T c / (0) > 3.5 U/zt 1: BEC of pars, (0) U, ξ(0) a 0, k B T c / (0) t 2 /U 2

28 Hghlght U = 2 (weak), U = 6 (ntermedate), U = 12 (strong) U = 2: already 2 (0)/k B T c 10, much greater than BCS! gap closes wth ncreasng T, band lke DOS for T > T c. U = 6: larger gap, 2 (0)/k B T c 20, larger T c dstnct pseudogap near T c, perssts to T 2T c the φ s have a broad dstrbuton at hgh T. U = 12: gap U, suppressed T c t 2 /U clean gap perssts to T T c. the thermal transton s from an nsulator to a SF

29 A further sgnature of correlatons above T c : The correlaton φ 0 φ cos(θ 0 θ ) at U = 2 and U = 12 at three dfferent T Even U = 2 has sgnfcant φ ampltudes above T c.. non BCS..

30 Dff regmes at U = 2, U = 6 and U = 12. All have SC ground states. Even at U = 2 parng ampltude survves at 2T C and form domans. Already outsde the BCS regme. No vsble pseudogap. The U = 6 case shows qualtatvely smlar physcs, but dstnct pseudogap, and strong par correlatons at the hghest T. U = 12 shows suppressed ampltude fluctuatons, clean gap n the DOS. Phase fluctuatons domnate.

31 C. Superflud- nsulator transtons H = t j σ (c σ c jσ + h.c.) + σ (ɛ µ)n σ U n n The -ve U model wth dsorder descrbes SF-nsulator transtons (SIT). ɛ dstrbuted between ±V. Small V : dsordered SF, large V : gapped nsulator. There s already a phase dagram based on a BdG + flucns theory V/t (a) (b) (c) Gapped Insulator 1.2 s-wave superconductor U /t Ghosal, et al., PRB (2001).

32 Illustratve result: Dsorder at weak couplng, low T. U=2,T=0.005 V=0 V=1 V=3 V=5 DOS E Loss of coherence peak n the DOS.. Can be captured wthn BdG as well (Ghosal..) But the thermal physcs?

33 Tc /Tc_0 vs. DIS, U=2 mod_conf_mc_2000_t0.006.txt u 2:3:1 Tc / Tc_ V x y Three benchmark results: () Suppresson of T c wth dsorder (left), albet slower than full QMC. () The presence of a pseudogap n the DOS even above T c (mddle). () The formaton of superconductng slands, n the dsordered system What s our advantage wth respect to (dsordered) BdG theory? The HS based approach s equv to BdG at low T, captures ampltude and phase flucns at fnte T.

34 Dsorder dependence of the DOS at ntermedate couplng. Promnent pseudogap even at low temperature. REN. Tc vs DIS,U = 4 REN. Tc vs. DIS., U=12 T_c / T_c V T_c / T_c V Dsorder suppresson of T c at ntermed and strong couplng.

35 D. Inhomogeneous superflud n a trap H = t (c σ c jσ + h.c.) + V 0 (x 2 + y 2 )n σ µ ˆN U j σ σ n n We explore U = 2, 6, 12, µ = U/2, and V b = U/2, U, 2U.

36 T dependence at strong nteracton U = 12, V b = 2U. Quaspartcle densty of states: T dep at strong couplng, U = 12, V b = 2U. the coherence feature at low T and ts T dependence the persstent gap n the spectrum above T c the confnement nduced oscllatons, T ndep

37 Concluson -Optcal lattces allow controllable realsaton of many body models. -They can be used as analog smulator for correlated systems. -May be used to study dynamcs and non-equlbrum effects as well. -The nhomogenety s an essental feature: novelty, dffculty.. -We have a method that handles strong couplng, dsorder and confnement. -Easy extenson to d-wave parng, FFLO states, vortex lattces... -More measurement tools, and theoretcal concepts need to be developed.. Huge lterature! two references.. Jaksch and Zoller, Ann. Phys., 315, 52 (2005) Bloch, Dalbard and Zwerger, Rev. Mod. Phys. 80, 885 (2008)