Metal-Mott Insulator Transition and Spin Exchange of Two-Component Fermi Gas with Spin-Orbit Coupling in Two-Dimension Square Optical Lattices
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] O c t Metal-Mott Insulator Transition and Spin Exchange ofTwo-Component Fermi Gas with Spin-Orbit Coupling inTwo-Dimension Square Optical Lattices
Beibing Huang ∗ Department of Experiment Teaching, Yancheng Institute of Technology, Yancheng, 224051, Chinaand Shaolong WanInstitute for Theoretical Physics and Department of Modern Physics,University of Science and Technology of China, Hefei, 230026, ChinaJuly 18, 2018
Abstract
Effects of spin-orbit coupling (SOC) on metal-Mott insulator transition(MMIT) and spin exchange physics (SEP) of two-component Fermi gasesin two-dimension half-filling square optical lattices are investigated. Inthe frame of Kotliar and Ruckenstein slave boson and the second orderperturbation theory, the phase boundary of paramagnetic MMIT andspin exchange Hamiltonian are calculated. In addition by adopting twomean-field ansatzs including antiferromagnetic, ferromagnetic and spiralphases, we find that SOC can drive a quantum phase transition fromantiferromagnet to spiral phase.
PACS number(s): 03.75.Ss, 51.60.+a, 05.70.Fh
In a crystalline solid spin-orbit coupling (SOC), which occurs naturally in systems withbroken inversion symmetry and makes the spin degree of freedom respond to its orbital mo-tion, is responsible for many interesting phenomena, such as magnetoelectric effect [1, 2, 3], ∗ Corresponding author. Electronic address: [email protected] K atoms [23], BCS-BEC crossover in the two-componentFermi gases with SOC have been widely studied [24, 25, 26, 27, 28, 29, 30, 31].By contrast in this paper we consider repulsive two-component Fermi gases with SOCin a two-dimensional square optical lattice, and are interested in the effects of SOC onmetal-Mott insulator transition (MMIT) and spin exchange physics (SEP) at half filling.Essentially MMIT of two-component Fermi gases without SOC in an optical lattice hasbeen realized experimentally [16] and is driven by the competition between hopping termand on-site interaction in the frame of one-band Hubbard model. When the hopping termdominates, atoms conduct freely in a lattice and the system is a metal. Gradually adjustingon-site interaction to an extent that the gain of the kinetic energy cannot offset the increaseof potential energy, on-site interaction forbids the hopping of atoms and the system evolvesinto Mott insulator (MI). In a MI, SEP is correctly described by quantum antiferromagneticHeisenberg model, which is known from the famous t − J model [32] expected to offer amechanism for high temperature superconductor. Physically this effective antiferromagneticcoupling between the nearest-neighbor spins comes from Pauli exclusion principle and thefact that the hopping of a particle cannot change its spin. Thus to minimize kinetic energythe nearest-neighbor spins must be antiparallel. In the presence of SOC, it has two effectson atom hopping. On the one hand SOC can make atoms move from one site to anothersite and corresponds to an effective hopping term, so it definitely has important effects onMMIT in view of the above statement. On the other hand, the effective hopping inducedby SOC is spin-flipped to support the nearest-neighbor spins parallel. From this viewpointSOC also dramatically changes SEP. Furthermore it is likely that when the strength ofSOC is beyond certain critical value, the system will show a quantum phase transition fromantiferromagnetic to other magnetic states.This paper is organized as follows. In section 2, we firstly derive Hubbard Hamiltonianin two-dimensional square optical lattices with SOC, then by using Kotliar-Ruckenstein(KR) slave bosons [33] we investigate paramagnetic MMIT, i.e. Brinkman-Rice phase tran-sition [34]. In section 3 under the limit of large on-site repulsion and using the secondorder perturbation theory spin exchange Hamiltonian is obtained. By making mean fieldapproximations we find that the ground state of the system is either antiferromagnetic orspiral depending on the relative magnitude of hopping term and strength of SOC and aquantum phase transition happens between them. The conclusions are given in section 4.2 Metal-MI Phase Transition with SOC
The Hamiltonian of the system we consider is H = Z d ~r X α,β Ψ † α ( ~r ) " ~p m + V OL ( ~r ) + λ ( σ x p y − σ y p x ) Ψ β ( ~r ) + g Ψ †↑ ( ~r )Ψ †↓ ( ~r )Ψ ↓ ( ~r )Ψ ↑ ( ~r ) , (1)where a Fermi atom of mass m for spin α is described by the field operators Ψ α ( ~r ) and V OL ( ~r ) is optical potential for two-dimensional square lattices. λ , g ( >
0) and ~σ representthe strength of Rashba SOC, two-body contact interaction and Pauli matrix respectively.When temperature is very low and filling factor is not too high, all atoms are constrainedinto the lowest band of the optical lattice. Expanding the field operator in terms of theWannier functions Ψ α ( ~r ) = P i a iα w ( ~r − ~R i ), where a iα is the annihilation operator foran atom of spin α in site ~R i , and only retaining on-site interaction and nearest neighborhopping, we find H = X a † iα t αβij a jβ + U X i a † i ↑ a † i ↓ a i ↓ a i ↑ , (2)where the hopping term t αβij is a 2 × t ↑↑ ij = t ↓↓ ij = − t , t ↑↓ ij = − h t ↓↑ ij i ∗ = Γ xi,j − i Γ yi,j . These parameters t , Γ xi,j , Γ yi,j and U are related to the Wannierfunction as follows t = − Z d ~rw ( ~r − ~R i ) " ~p m + V OL ( ~r ) w ( ~r − ~R j ) ,U = g Z d ~rw ( ~r − ~R i ) w ( ~r − ~R i ) w ( ~r − ~R i ) w ( ~r − ~R i ) , Γ xi,j = λ Z d ~rw ( ~r − ~R i ) ∂∂x w ( ~r − ~R j ) , Γ yi,j = λ Z d ~rw ( ~r − ~R i ) ∂∂y w ( ~r − ~R j ) . (3)From above expressions (3) and the symmetry of Wannier function, Γ xi,j , Γ yi,j satisfy therelations Γ xi,j = Γ yi,j = − Γ xj,i = − Γ yj,i . Moreover Γ xi,j = 0 if i, j are nearest neighbor along y direction and Γ yi,j = 0 if i, j are nearest neighbor along x direction. For convenience theparameter Γ is defined Γ = | Γ xi,j | = | Γ yi,j | to represent the strength of SOC.It is well known that MMIT is a phenomenon of strong correlation. In terms of strongcorrelation, apart from some numerical methods, such as dynamical mean-field theory [35],a few analytical methods are also available. The first is Gutzwiller variational wave function[36]. In this method to make the calculation tractable, one has to introduce the Gutzwillerapproximation which is basically at the mean-field level. Although this method is successfulto predict the existence of MMIT, it still has some disadvantages from variational andmean-field approximations. Another method is KR slave bosons [33]. It exactly reproducesthe results of Gutzwiller approximation at the saddle-point level and can be improvedsystematically by considering fluctuations around the saddle point [37]. Hence below weadopt slave bosons to study MMIT with SOC, although our results is at the saddle-pointlevel. 3or two-component Fermi gases, the Hilbert space for every lattice site i consists of fourstates | > i , | α > i = a † iα | > i and | ↑ , ↓ > i = a † i ↑ a † i ↓ | > i . In the representation of KR slavebosons, in addition to original fermions, a set of four bosons e , d , p α for every lattice siteare introduced so that | > i = e † i | vac > , | α > i = p † iα a † iα | vac > i and | ↑ , ↓ > i = d † i a † i ↑ a † i ↓ | vac > i ,where | vac > is the vacuum state after introducing slave bosons. It is easily found that e † i e i , d † i d i and p † iα p iα represent the projectors on the empty, doubly occupied and singlyoccupied site. Due to the fact that the introduction of bosons enlarges the Hilbert spaceof every site to contain some unphysical states, such as e † i e † i | vac > i etc., we must imposethree constraints e † i e i + p † iα p iα + d † i d i = 1, a † iα a iα = p † iα p iα + d † i d i . In terms of these bosonsand considering above constraints the Hamiltonian (2) is reformulated into H = X a † iα z † iα t αβij z jβ a jβ + U X i d † i d i , (4)with z iα = (1 − d † i d i − p † iα p iα ) − / z iα (1 − e † i e i − p † i − α p i − α ) − / ,z iα = e † i p iα + p † i − α d i . (5)As claimed by KR, the substitution z iα for z iα ensures z † iα z jβ = 1 to recover the results inthe limit U = 0 at the saddle-point approximation.The partition function Z can be written as a functional integral over the fermion andboson operators Z = Z D a α D e D p α D d Y iσ dλ (1) i dλ (2) iα exp [ − Z β L ( τ ) dτ ] , (6)where the Lagrangian L ( τ ) is L ( τ ) = X i e † i [ ∂∂τ + λ (1) i ] e i + d † i [ ∂∂τ + U + λ (1) i − λ (2) i ↑ − λ (2) i ↓ ] d i + p † iα [ ∂∂τ + λ (1) i − λ (2) iα ] p iα + X a † iα (cid:20) ( ∂∂τ + λ (2) iα − µ ) δ αβ δ ij + z † iα t αβij z jβ (cid:21) a jβ − λ (1) i , (7)and µ , λ (1) i , λ (2) iα are the chemical potential and Lagrange multipliers, respectively.Assuming uniform and static boson operators and Lagrange multipliers, i.e. at the saddlepoint, one can integrate over fermion operators and obtain for thermodynamic potential ofsingle siteΩ = λ (1) ( e + d + p α −
1) +
U d − λ (2) α ( d + p α ) − βN X kα ln [1 + e − βE kα ] (8)with E kα = h ǫ k ↑ + ǫ k ↓ + α q ( ǫ k ↑ − ǫ k ↓ ) + 4 z ↑ z ↓ Γ k i / ǫ kα = ǫ k z α − µ + λ (2) α , ǫ k = − t (cos k x a +cos k y a ), Γ k = 2Γ q sin k x a + sin k y a . It is to be noted that a , N are lattice length andthe number of lattice site, and wavevector k belongs to two dimensional Brillouin zone. At4his time the seven parameters e , p α , d , λ (1) and λ (2) α are obtained by minimizing Ω, andthe chemical potential at half filling by thermodynamic relation − ∂ Ω ∂µ = 1. These equationsare called saddle-point and number equations.From ∂ Ω ∂λ (1) = ∂ Ω ∂λ (2) α = 0 and − ∂ Ω ∂µ = 1, one can get e = d . Supposing paramagneticsolution p ↑ = p ↓ , then p = − d , z ↑ = z ↓ = z = 8 d (1 − d ). According to ∂ Ω ∂p α = ∂ Ω ∂e = ∂ Ω ∂d = 0, λ (2) ↑ = λ (2) ↓ = U , ǫ k ↑ = ǫ k ↓ , E kα = [ ǫ k + α Γ k ] z − µ + U , λ (1) = U − ξd (3 − d )with ξ = 1 N X k (cid:20) ǫ k + Γ k e βE k ↑ + 1 + ǫ k − Γ k e βE k ↓ + 1 (cid:21) . (9)Substituting above relations into saddle-point and number equations, one still has twoequations satisfied by µ and d N X k (cid:20) e βE k ↑ + 1 + 1 e βE k ↓ + 1 (cid:21) = 1 ,U + 8 ξ (1 − d ) = 0 . (10)At zero temperature, in the frame of KR slave bosons, d = 0 corresponds to thevanishing of the number of doubly occupied sites and indicates that the system is undergoinga MMIT. From this criterion one has numerically solved the equations (10). The numericalresults suggest the chemical potential is still fixed at µ = U/
2. Hence at zero temperature ξ = 1 N X k { ( ǫ k + Γ k )Θ[ − ( ǫ k + Γ k )] + ( ǫ k − Γ k )Θ[ − ( ǫ k − Γ k )] } . (11)and the phase boundary of MMIT is U = − ξ, (12)where Θ( x ) is Heaviside step function. Without SOC, ξ = N P k ǫ k Θ[ − ǫ k ] and the phaseboundary (12) is the same as the result in [33]. In Fig.1 the phase boundary of MMITis shown. From Fig.1, very explicitly SOC stabilizes the MI, which is consistent with thefact that SOC can be regarded as an effective hopping term. Besides instead of adjusting t MMIT can also be driven by changing SOC, so one has found another way to realize theMMIT.
As demonstrated in the section 2, at half filling when
U >> t and
U >>
Γ, the hopping ofatoms are forbidden and the system evolves into MI with spin S = for every lattice site.In the MI we could regard t and Γ as perturbations. In the limit of t = Γ = 0 the energy ofthe system does not depend on the spin orientations on different sites. When t , Γ are finitebut small, we expect that we still have spin S = in each site, but atom hopping processesinduce effective interactions between these spins, usually called spin exchange interaction[38]. To construct an effective spin exchange Hamiltonian for this system, we note that5n the second order in t , Γ it can be written as a sum of interaction terms for all nearestneighbor sites. These pairwise interactions can be found by solving a two-site problem inthe second order in t , Γ.The ground state manifold for two-site problem with one atom in each site composes offour degenerate zero-energy states | > = | ↑ > i | ↑ > j , | > = | ↑ > i | ↓ > j , | > = | ↓ > i | ↑ > j , | > = | ↓ > i | ↓ > j , (13)with i, j labelling two sites. The first order perturbation theory takes us out of the groundstate manifold and can be neglected. In the second order atom hoppings can connect allfour states by two intermediate states | > = | ↑ , ↓ > i | > j and | > = | > i | ↑ , ↓ > j . Tofind spin exchange Hamiltonian we need calculate all matrix elements [38] M ab = X c < a | H k | c >< c | H k | b >E b − E c , (14)where states | a >, | b > and | c > respectively belong to ground state manifold and in-termediate states with E representing eigenenergy of corresponding state in the zerothorder. The calculation is very direct and when i, j are nearest neighbor along x direc-tion, we have M = M = M = M = xij Γ xji U , M = M = M = M = t Γ xji U , M = M = M = M = t Γ xij U , M = − M = − M = M = − t U . Accord-ing to spectral representation of an operator, magnetic Hamiltonian of two-site problem is H i,j = P a,b | a > M ab < b | . Making substitutions | > → a † i ↑ a † j ↑ , | > → a † i ↑ a † j ↓ , | > → a † i ↓ a † j ↑ , | > → a † i ↓ a † j ↓ and using algebra of spin operator ~S i = a † iα ~σ αβ a iβ , we get H i,j = 4 t U ~S i · ~S j + 8 t Γ xi,j U ( S ix S jz − S iz S jx ) + 4Γ xi,j Γ xj,i U ( S iz S jz + S ix S jx − S iy S jy ) . (15)By the same procedure, when i, j are nearest neighbor along y direction we get H i,j = 4 t U ~S i · ~S j + 8 t Γ yi,j U ( S iy S jz − S iz S jy ) + 4Γ yi,j Γ yj,i U ( S iz S jz + S iy S jy − S ix S jx ) . (16)Thus spin exchange Hamiltonian of the whole system is H se = X H i,j . (17)Some comments about (17) is following. If Γ = 0 H se describes isotropic quantumantiferromagnet, consistent with t − J model. When Γ = 0, main effect of SOC is to breakspin conservation by two ways, one of which, corresponding to the second term in (15) and(16), flips one spin of two nearest neighbor sites, while the other flips simultaneously twospins corresponding to the third term in (15) and (16). Thus the antiferromagnetic statewill be unstable when the strength of SOC Γ is beyond certain critical value.Now we decide the ground state of the system at mean-field level. This corresponds toregard quantum spin operator ~S i as a classical vector. The first mean-field ansatz including6erromagnetic and antiferromagnetic states is that spin configurations in two sublattices ofa square lattice take different values specified respectively by coordinate angle ( ϑ, ϕ ) and( γ, δ ) with 0 ≤ ϑ, γ < π and 0 ≤ ϕ, δ < π , the mean-field energy scaled by U is E = 8 N h ( e t − e Γ ) cos ϑ cos γ + e t sin ϑ sin γ cos( ϕ − δ ) i , (18)where a 2 N × N lattice is assumed and e t = t/U , e Γ = Γ /U . Easily found that energy onlydepends on the difference of ϕ and δ , for convenience we can choose δ = 0. Owing to factorsin ϑ sin γ ≥
0, the minimization of energy leads to ϕ = π . Minimizing energy about ϑ, γ ,we get ( e t − e Γ ) sin ϑ cos γ + e t cos ϑ sin γ = 0 , ( e t − e Γ ) cos ϑ sin γ + e t sin ϑ cos γ = 0 . (19)Equations (19) have two sets of solution ϑ = γ = 0 and ϑ = γ = π/
2. The first solutioncorresponds to ferromagnet along z direction with E F E = 8 N ( e t − e Γ ) and the secondcorresponds to antiferromagnet along x direction with E AF = − N e t . If E F E < E AF ground state is ferromagnetic, on the contrary ground state is antiferromagnetic. Thus thismean-field ansatz predicts a phase transition from antiferromagnet to ferromagnet and thecritical point is E F E = E AF , i.e. e Γ = √ e t .The motivation of the second mean-field ansatz comes from t = 0 limit in spin exchangeHamiltonian (17). Letting t = 0 the classical spin configuration minimizing energy satisfiesthree conditions: (1) z components of all spins are equal; (2) for a random chain along x direction x components of all spins are equal but y component must be alternating; (3) fora random chain along y direction y components of all spins are equal but x component mustbe alternating. Such spin configuration, which we call spiral phase and shown in Fig.2, ispermissible in a square lattice. From above three conditions if coordinate angle ( θ, φ ) of aspin in the lattice is specified, energy of the system is E SP = 8 N ( e t cos θ − e Γ ) , (20)and its minimization gives rise to θ = π/ E SP = − N e Γ . Comparing E SP with E F E we find that the ferromagnetic state is always a metastable state. As a result phase tran-sition predicted by the first mean-field ansatz does not exist, we get a phase transitionfrom antiferromagnet to spiral phase with critical point e Γ = e t . Fig.1 also shows magneticphase diagram in terms of such two mean-field ansatzs. Physically the metastability offerromagnetic state is attributed to the fact that SOC breaks spin conservation. In conclusion we have discussed MMIT and SEP of two-component Fermi gases with SOCin two-dimensional half-filling square optical lattices in the frame of KR slave bosons andsecond-order perturbation theory. Comparing with the case without SOC, SOC not onlyenlarges the region of MI in the phase diagram and introduces another way to realize MMIT,but also dramatically affects SEP due to SOC breaking spin conservation. Importantlyby adopting two mean-field ansatzs we find that SOC can drive a phase transition from7ntiferromagnet to spiral phase. Experimentally this phase transition can be observed byeither adjusting optical lattices to suppress the hopping term or decreasing the strength ofSOC.
Acknowledgement
The work was supported by National Natural Science Foundation of China under Grant No.10675108. The author Huang also thanks Foundation of Yancheng Institute of Technologyunder Grant No. XKR2010007.
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Figure 2: Spin configuration of spiral phase is shown with small squares representing latticesites. ( θ, φ ), ( θ, − φ ), ( θ, π − φ ) and ( θ, π + φφ