General Hubbard model for strongly interacting fermions in an optical lattice and its phase detection
aa r X i v : . [ c ond - m a t . o t h e r] J un General Hubbard model for strongly interacting fermions in an optical latticeand its phase detection
L.-M. Duan
FOCUS center and MCTP, Department of Physics, University of Michigan, Ann Arbor, MI 48109
Based on consideration of the system symmetry and its Hilbert space, we show that strongly inter-acting fermions in an optical lattice or superlattice can be generically described by a lattice resonanceHamiltonian. The latter can be mapped to a general Hubbard model with particle assisted tunnelingrates. We investigate the model under population imbalance and show the attractive and the repul-sive models have the same complexity in phase diagram under the particle-hole mapping. Using thismapping, we propose an experimental method to detect possible exotic superfluid/magnetic phasesfor this system.
PACS numbers: 03.75.Ss, 05.30.Fk, 34.50.-s
Among the control techniques for ultracold atoms, op-tical lattice and Feshbach resonance play particularly im-portant roles. The optical lattice is used to control theinteraction configuration while the Feshbach resonanceis a tool to tune the interaction magnitude. The com-bination of these two powerful techniques naturally be-comes the next frontier, which has attracted significantrecent interest [1, 2, 3, 4, 5]. To understand this im-portant system, one needs to have a Hamiltonian to de-scribe strongly interacting atoms in an optical lattice.The starting Hamiltonian is unfortunately complicatedas one has to take into account multi-band populationsas well as direct neighboring couplings [2, 4, 5]. We havedescribed a method in [5] to derive an effective latticeHamiltonian for this system from field theory of the two-channel model.In this paper, we report the following advance alongthis direction: firstly, based on consideration of the sys-tem symmetry and its Hilbert space, we show that a lat-tice resonance model turns out to be a generic Hamilto-nian for this system. The resulting Hamiltonian agreeswith the one from our previous microscopic derivation [5],but the method used here shows this Hamiltonian shouldhave general applicability. As an example, we point outthat for strongly interacting fermions in optical superlat-tices, the effective Hamiltonian is again described by thislattice resonance model when we introduce some dresseddegrees of freedom. For certain configurations of the su-perlattice, the system naturally supports d-wave super-fluid. Secondly, we mathematically map the lattice reso-nance Hamiltonian to a general Hubbard model (GHM)with particle assisted tunneling rates. The particle as-sisted tunneling brings in some new feature, in particu-lar, it may favor a superfluid phase compared with theHubbard model. Thirdly, we discuss the attractive Hub-bard model with population imbalance between the twospin components, and show it has the same complexityin phase diagram as the repulsive Hubbard model undera particle-hole mapping. This result is related to the re-cent large effort to understand the polarized fermi gas [6, 7]. Finally, using the mapping above, we propose anexperimental scheme to detect possible exotic superfluidor magnetic orders in this system. The method is basedon Raman-pulse-assisted time-of-flight imaging, and canreveal the superfluid or magnetic phases with detailed in-formation about the order parameter or the pairing wavefunction.For strongly interacting two-component (effectivelyspin-1 /
2) fermions in an optical lattice, when two atomswith different spins come to the same site, they form adressed molecule with atomic population distributed overmany lattice bands due to the strong on-site interaction[4, 5]. We consider the system with an average atom fill-ing number n ≤
2. In this case, we can neglect the 3-atomoccupation of a single site as that is suppressed at lowtemperature by an energy cost about the lattice band gap[8]. We then have only four possible configurations foreach site i , either empty, or a spin- σ ( σ = ↑ , ↓ ) atom, or adressed molecule. The creation operators for these con-figurations are denoted by b † i , a † iσ , d † i , respectively, whilethe corresponding states are written as | b i i , |↑ , ↓i i , | d i i .We introduce the slave boson operator b † i for an emptysite i so that the constraint of the Hilbert space on eachsite can be simply implemented through b † i b i + a † i ↑ a i ↑ + a † i ↓ a i ↓ + d † i d i = I. (1)Note that with this constraint, a iσ describe fermionswhile d i and b i both represent hard-core bosons.We assume the system has a global SU(2) symmetryfor the spin components. In that case, |↑i i and |↓i i are degenerate in energy, and the most general formof the single-site Hamiltonian can be written as H i = − µ P σ a † iσ a iσ + (∆ − µ ) d † i d i , where we have absorbedthe single-atom energy into the definition of the chemi-cal potential µ , and ∆ is the relative energy shift of thedressed molecule. For two neighboring sites i and j , dueto the atomic tunneling and off-site interactions, therewill be a Hamiltonian term H ij to describe all the possi-ble configuration tunneling or couplings. With the spinSU(2) symmetry and the number conservation of eachspin component, the most general two-site Hamiltoniancan be written as H ij = H (1) ij + H (2) ij , where H (1) ij de-scribes the configuration tunneling that involves transferof one atom with the following form (see the illustrationin Fig. 1) H (1) ij = X σ (cid:16) ta † iσ b i b † j a jσ + t da d † i a iσ a † jσ d j (cid:17) + g ( d † i b j + d † j b i )( a i ↑ a j ↓ − a i ↓ a j ↑ ) + H.c., (2)and H (2) ij describes the configuration coupling that in-volves real or virtual tunneling of two atoms with thegeneral expression H (2) ij = (cid:16) t d d † i b i b † j d j + H.c. (cid:17) + x d n di n dj + x a n i n j + x s s i · s j + x b n bi n bj . (3)In H (2) ij , the number and the spin operators are de-fined by n di ≡ d † i d i , n i ≡ a † i ↑ a i ↑ + a † i ↓ a i ↓ , n bi ≡ b † i b i ,and s i ≡ P σσ ′ a † iσ σ σσ ′ a iσ ′ / σ σσ ′ is the Pauli ma-trix). The term n bi n bj is equivalent to the cross cou-pling n di n j + n i n dj under the constraint (1). By analyz-ing the level configurations in Fig. 1, one can convinceoneself that H (1) ij and H (2) ij include all the possible two-site coupling terms with the SU(2) symmetry. As theatomic interactions are short-range, all the multiple sitecouplings can be neglected. So a generic lattice Hamil-tonian is given by H = P i H i + P h i,j i (cid:16) H (1) ij + H (2) ij (cid:17) ,where h i, j i denotes neighboring sites. This Hamiltoniandescribes the coupling between the fermionic atoms a iσ and the bosonic dressed molecules d i with a detuning ∆,and will be referred in the following as the lattice reso-nance model. ¯ db gt da t ¯ db g t da t FIG. 1: Illustration of the configuration tunnelling betweentwo neighing sites. The process shown in the figure correspondto the t , g , and t da terms in the Hamiltonian. The Hamiltonian H , together with the constraint (1),poses a well-defined problem. Note that H agrees in formwith the effective lattice Hamiltonian for strongly inter-acting fermions that we derived before from a completelydifferent method [5]. The only specification from thatmicroscopic derivation is to fix the coefficients x b = 0and x a = − x s /
4. As mentioned in [5], in the case of alarge detuning ∆, the Hamilton H is reduced to either the t-J model for atoms or the XXZ model for dressedmolecules, depending on which species get populated.We also notice that for short range interactions, withincrease of the lattice potential barrier, all the interac-tion coefficients in H (2) ij decay much faster compared withthose in H (1) ij . So for a lattice with sufficient depth, H (1) ij dominates over H (2) ij , and in the following, without spe-cial mention we will consider the simplified Hamiltonian H = P i H i + P h i,j i H (1) ij by dropping H (2) ij .We now recast the Hamilton H into a different formwhich shows its connection with the Hubbard model. Forthis purpose, we map the dressed molecule state d † i | vac i to the two-fermion state a † i ↓ a † i ↑ | vac i , where | vac i denotesthe vacuum. Note that physically the structure of thedressed molecules should be determined by diagonaliz-ing the on-site interaction Hamiltonian, and it generallyinvolves superposition of atoms in many band configu-rations [4, 5], which is certainly different from the state a † i ↑ a † i ↓ | vac i with double occupation on a single band. Butmathematically we can identify these two states by a one-to-one mapping. After this mapping, the Hamiltonian H can be written in the form H = X i [(∆ / n i ( n i − − µn i ] (4)+ X h i,j i ,σ [ t + δg ( n iσ + n jσ ) + δtn iσ n jσ ] a † iσ a jσ + H.c. where δg ≡ g − t , δt ≡ t da + t − g and n iσ ≡ a † iσ a iσ ( σ = ↓ , ↑ for σ = ↑ , ↓ ). To verify the two forms of H inEqs. (2) and (4) are equivalent to each other, one cancheck the physical process represented by each term toconfirm it is identical. Note that in this new form of H ,there is no need of the slave boson operator to constraintthe Hilbert space as the latter is automatically fixed bythe properties of fermions. As there is no additional con-straint, the Hamiltonian in the form of Eq. (4) lookssimpler and may be easier for treatment in certain cases.Compared with the conventional Hubbard model, the ef-fective tunneling rate in H becomes an operator whichdepends on occupation of the two sites. The original lat-tice resonance Hamiltonian in Eq. (2) is thus mappedto a general Hubbard model ((GHM) with particle as-sisted tunneling rates. For weakly interacting fermions,the multi-band population and the direct neighboringcoupling become negligible, then the coefficients g and t da tend to t a , and the GHM returns to the conventionalHubbard model as one expects in this case [9].The derivation of the Hamiltonian H in this work isbased on very general arguments about the single-siteHilbert space and the system symmetry. This remindsus that H has a generic form which should apply to dif-ferent systems with similar Hilbert space structure andsymmetry properties. As an example, we point out thatfor interacting fermions in an optical superlattice, un-der several interesting configurations, the system is alsowell described by the above Hamiltonian H . Figure 2Aillustrates an optical superlattice potential which canbe realized with two standing wave laser beams [10].With a combination of this superlattice and the con-ventional optical lattice potentials, one can realize thedimer or plaquette lattices as illustrated in Fig. 2B and2C where the intra-dimer (intra-plaquette) couplings aremuch stronger than the inter-dimer (inter-plaquette) cou-plings. To derive an effective Hamiltonian for this sys-tem, one needs to first construct dress energy levels foreach dimer (plaquette) by exactly solving a few-site prob-lem. For two-component interacting fermions in thoselattices near half filling, the low energy level configura-tions from each dimer (plaquette) have basically the samestructure as those shown in Fig. 1 [11, 12, 13], and thesystem also has the SU(2) symmetry. We then immedi-ately conclude that the Hamiltonian in the forms of Eq.(2) or (4) should be applicable to describe physics in thedimer or plaquette lattices around half filling. The two-dimensional plaquette lattice is particularly interesting:because of the internal plaquette structure, the excitationfrom | b i i to | d i i states in Eq. (2) has a d -wave symmetry(e.g., D d † i b i E flips sign under a π/ d -wave superfluid [11, 12]. AB C
FIG. 2: Illustration of an optical superlattice: (A) The super-lattice potential. (B,C) The dimer and the plaquette lattices(bold lines represent stronger coupling) formed with the po-tential in (A).
We now investigate some properties of the GHM inEq. (4). When the detuning ∆ is negative, similar to theHubbard model, we expect this Hamiltonian is in a super-fluid state away from the unit filling. When ∆ is positive,although it is not clear yet whether H has a superfluidstate, compared with the corresponding repulsive Hub-bard model, we do expect that the superfluid possibilitybecomes higher when g > t a (which is very likely thecase for fermionic atoms near a wide Feshbach resonance[5]) as a large g term (see Eq. (2)) clearly favors Cooperpairing. From the single-site physics, we know that twoatoms always have an on-site bound state (correspondingto a negative ∆) with the binding energy ( − ∆) approach-ing zero as one moves to the BCS side of the Feshbach resonance [4, 5]. So in the ground-state configuration,the strongly interacting fermi gas naturally implementthe GHM with a negative ∆. To experimentally investi-gate the GHM with a positive ∆, one needs to start withthe population in atoms (instead of Feshbach molecules),and to approach the Feshbach resonance from the BECside (the system is in a metastable state in this case).The effective Hamiltonians in different regions are shownin Fig. 3. t-J XXZ AGHMRGHMBEC BCS
FIG. 3: The effective Hamiltonians in different regions. Thesolid curved correspond to two dressed molecule bands, andthe middle dashed line is an atomic band. On the BCS orBEC (with population mainly in atoms) sides, the Hamitoni-ans are given by the attractive (repulsive) general Hubbardmodels (AGHM and RGHM), respectively. As one increasesthe detuning ∆, one gets either the XXZ model for the dressedmolecules or the t-J model for the atoms. In the deep BCSor BEC limit, the GHM returns to the attractive or repul-sive Hubbard model, and the XXZ Hamiltonian yields to thebosonic Hubbard model for molecules when multiple occupa-tion of a single site is allowed by weaker effective interaction.
When we take into account possible population im-balance between the two spin components, the repulsiveand the attractive GHMs (with positive or negative ∆,respectively) become intrinsically connected, and theyshould have the same complexity in phase diagram. Po-larized fermi gas recently raised a lot of interest [6], andin free space (or in a weak trap), although population im-balance yields some new features, the basic physics thereis still largely captured by an extension of BCS type ofmean-field theory [7]. However, for polarized fermi gasin an optical lattice, we show that simple extensions ofthe BCS theory are very likely to give misleading resultsbecause of the exact mapping between the repulsive andthe negative GHMs. Population imbalance correspondsto introduction of an effective magnetic field h , whichadds a term − h P i σ zi ( σ zi ≡ n i ↑ − n i ↓ ) to the Hamilto-nian H in Eq. (4). We apply a particle-hole transforma-tion a i ↑ → a i ↑ and a i ↓ → ( − i a † i ↓ to the Hamiltonian[14] (for simplicity, we consider a bi-partite lattice). Un-der this transformation, the Hamiltonian H − h P i σ zi ismapped to H ′ = X i [ − (∆ / n i ( n i − − µ ′ σ zi − h ′ n i ] (5)+ X h i,j i ,σ [ t σ + δg σ ( n iσ + n jσ ) + δtn iσ n jσ ] a † iσ a jσ + H.c. where the parameters µ ′ ≡ µ − ∆ / h ′ ≡ h − ∆ / t ↑ ≡ t , t ↓ ≡ t da , δg σ ≡ g − t σ , and we neglect the con-stant energy per site h − µ . One can see that an attrac-tive GHM (∆ <
0) is mapped exactly to a dual repulsivemodel (with − ∆), where the chemical potential µ and thefiled h exchange their roles. Superfluid phases (includingboth the BCS state and the LOFF (Larkin-Ovchinnikov-Fulde-Ferrel) state with pairing at nonzero momenta [15])of the original model correspond to magnetic phases ofthe dual model and vice versa. For a repulsive Hub-bard model on a square lattice, the magnetic order existsonly in a region near half filling, and with hole dopingthere are possibilities of exotic phases including a non-BCS superfluid state. This suggests for the attractiveHubbard model with population imbalance, the super-fluid phase exists in the region with small polarization.With further increase of the polarization, there could ap-pear exotic phases including a d-wave magnetic order.Experimental investigation of the attractive GHM withpopulation imbalance (which might be easier for realiza-tion compared with the repulsive one) is therefore able toprovide critical information to understand the challeng-ing phase diagram of the repulsive Hubbard model.Finally, we propose a method to detect possible exoticphases in this system by making use of the above map-ping. Our purpose is to directly measure the magnetic orsuperfluid order parameters. The detection scheme com-bines the time-of-flight imaging with some instantaneousRaman pulses [16]. We take Li atoms as a typical ex-ample. The scheme is illustrated in Fig. 4. Right afterturn-off of the trap, we immediately apply two consecu-tive impulsive Raman pulses. These pulses are assumedfaster than the system dynamics (characterized by theFermi energy), but slower compared with the level split-ting between the |↑i and |↓i levels (about 70 MHz). Thefirst is a π -pulse, consisting of two laser beams propagat-ing along different directions, which transfers the atomsfrom the level |↑i to | i by imprinting a photon recoilmomentum − q . As the level | i is detuned from |↑i bya few GHz, this transition at the same time tune thesystem out of Feshbach resonance (the atoms in states | i and |↓i are only weakly interacting) [16]. The secondis a π/ | i and |↓i , which induces a transformation a k → ( a k + e iϕ a k ↓ ) / √ a k ↓ → ( a k ↓ − e − iϕ a k ) / √ k ( ϕ is the relative laserphase). After these two pulses, we take the time-of-flightimages (with basically ballistic expansion) for the atomsin levels | i and |↓i , and the difference of these two im-ages give exactly the cross correlation of the ↑ and ↓ spin-components at different momenta: n k − n k ↓ = 2 Re (cid:16) e iϕ a † k + q , ↑ a k ↓ (cid:17) . (6)We now show through a few examples that we candirectly confirm various magnetic or superfluid phases with this detection ability. (i) For magnetic phaseswith a pretty general form of the spin order parameter h s i i = v cos( Q · r i ) + v sin( Q · r i ) [14], we can con-firm it with sharp peaks for the correlation in Eq. (6)when the relative momentum q is scanned to ± Q . Thespin vectors v and v can be inferred from the rela-tive laser phase ϕ . (ii) For the LOFF superfluid statewith pairing at a non-zero momentum q , the order pa-rameter h a k + q , ↑ a − k ↓ i is nonzero. After the particle-holemapping, this order parameter corresponds to a magneticorder D a †− k − q , ↑ a − k ↓ E of the dual Hamiltonian. The peakof the correlation function in Eq. (6) at the relative mo-mentum − q thus confirms Bose condensation to a non-zero pair momentum for the original Hamiltonian, andthe distribution in k of the correlation D a †− k − q , ↑ a − k ↓ E gives the original pair wavefunction. (iii) Similar to theLOFF state, for a d-wave superfluid phase with the or-der parameter h a k , ↑ a − k ↓ i ∝ cos k x − cos k y , , the pairwavefunction and its spatial symmetry can be directlymeasured by detecting the correlation (6) for the dualHamiltonian. ¯ p ¯ p Pulse I Pulse II
FIG. 4: Illustration of the two Raman pulses (a π and a π/ Li atoms as an example (with the magnetic field near thewide Feshbach resonance).
In summary, we have established the results as we out-lined in the introduction.This work was supported by the MURI, the DARPA,the NSF award (0431476), the DTO under ARO con-tracts, and the A. P. Sloan Fellowship. [1] J. K. Chin et al., Nature London 443, 961 (2006).[2] T. Stoeferle et al., Phys. Rev. Lett. 96, 030401 (2006).[3] D. B. M. Dickerscheid et al., Phys. Rev. A 71, 043604(2005); L. Carr and M. Holland, Phys. Rev. A 72, 031604(2005); F. Zhou and C. Wu, New J. Phys. 8, 166 (2006).[4] R. B. Diener and T.-L. Ho, Phys. Rev. Lett. 96, 010402(2006).[5] L.-M. Duan, cond-mat/0508745, Phys. Rev. Lett. 95,243202 (2005).[6] M. W. Zwierlein, A. Schirotzek, C. H. Schunck, and W.Ketterle, Science 311, 492 (2006); G. B. Partridge et al.,Science 311, 503 (2006). [7] For a review, see D. E. Sheehy, L. Radzihovsky,cond-mat/0607803.[8] J. Kestner, L.-M. Duan, unpublished.[9] W. Hofstetter et al., Phys. Rev. Lett. 89, 220407 (2002).[10] J. Sebby-Strabley, M. Anderlini, P. S. Jessen, J. V. Porto,cond-mat/0602103; S. Trebst, U. Schollwoeck, M. Troyer,P. Zoller, Phys. Rev. Lett. 96, 250402 (2006)..[11] W.-F. Tsai, S. A. Kivelson, Phys. Rev. B 73, 214510(2006).[12] T. Goodman and L.-M. Duan, Phys. Rev. A 74, 052711(2006) and unpublished. [13] For a plaquette configuration, the levels |↑i and |↓i|↓i