Spin-Orbit Driven Transitions Between Mott Insulators and Finite Momentum Superfluids of Bosons in Optical Lattices
Mi Yan, Yinyin Qian, Hoi-Yin Hui, Ming Gong, Chuanwei Zhang, Vito W. Scarola
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] N ov Spin-Orbit Driven Transitions Between Mott Insulators and Finite MomentumSuperfluids of Bosons in Optical Lattices
Mi Yan, Yinyin Qian, Hoi-Yin Hui, Ming Gong,
3, 4, 2
Chuanwei Zhang, ∗ and V.W. Scarola Department of Physics, Virginia Tech, Blacksburg, Virginia 24061 USA Department of Physics, The University of Texas at Dallas, Richardson, Texas 75080 USA Key Lab of Quantum Information, CAS, University of Science and Technology of China, Hefei, 230026, P.R. China Synergetic Innovation Center of Quantum Information and Quantum Physics,University of Science and Technology of China, Hefei, 230026, P.R. China
Synthetic spin-orbit coupling in ultracold atomic gases can be taken to extremes rarely found insolids. We study a two dimensional Hubbard model of bosons in an optical lattice in the presenceof spin-orbit coupling strong enough to drive direct transitions from Mott insulators to superfluids.Here we find phase-modulated superfluids with finite momentum that are generated entirely byspin-orbit coupling. We investigate the rich phase patterns of the superfluids, which may be directlyprobed using time-of-flight imaging of the spin-dependent momentum distribution.
PACS numbers: 03.75.Mn, 67.85.Hj, 67.25.dj
I. INTRODUCTION
The Rashba effect [1] in solids derives from the motionof an electron in a strong electric field. As the electronmoves in the presence of a potential gradient, ∇ V , itexperiences an effective magnetic field in its frame of ref-erence. The Rashba energy [1]:( ~ ∇ V × ~p ) · ~σ, (1)captures the energetics of electron spin reorientation dueto the effective magnetic field, where ~p is the particle mo-mentum and ~σ are the Pauli matrices. The Rashba spin-orbit coupling (SOC) energy is well known to be partic-ularly strong at metallic surfaces [2, 3] (e.g., on Ag(111)or Au(111)) because here we find extremely strong po-tential gradients. As a result, studies of the impact ofRashba SOC on two-dimensional (2D) conductors have along history [4]. But the impact of Rashba SOC on thesurface states of Mott insulators has come under morecareful scrutiny recently because of possible connectionsto topological insulators [5, 6].Mott insulators localize as a result of strong interactionand would therefore appear to exclude the possibility ofSOC effects, but one can argue that this is not alwaysthe case. Small momentum in Eq. (1) (the case for local-ized states) does not necessarily imply low Rashba ener-gies. In an extreme limit, Mott insulating surfaces can, inprinciple, experience very large potential gradients thatcan compensate the small momentum, i.e., h p i → h ~ ∇ V × ~p i ∼ E F , where E F is the Fermi energy. If, inthis limit, the energetics of Rashba SOC compete withthe Mott gap, one could observe a transition between aMott insulator and a conducting state driven entirely byRashba SOC in spite of the small average momentumof particles in Mott insulators. Unfortunately, the limit ∗ Electronic address: [email protected] where Rashba SOC competes with the Mott gap is rarein solids because it would typically be precluded by othereffects, such as charge transfer between bands. But thislimit can be explored in another context: using syntheticSOC in optical lattices.Recent experimental progress [7–13] demonstrates en-gineering of synthetic SOC for ultracold atomic gases[14]. These experiments show that Raman beams canbe used to dress atoms with a spin-dependent momen-tum. Rashba (and/or Dresselhaus) SOCs governing thesedressed states [15, 16] are tunable to extremes not possi-ble in solids, see Fig. 1. Recent work shows, for example,that synthetic SOC can generate flat bands [17–20], ex-otic superfluidity [21], and intriguing vortex structures[16, 22, 23].Recent theory work has also explored the impact ofSOC on the spin structure of Mott insulators in opticallattices [24–28]. Here super-exchange coupling betweensites was shown to combine with Rashba SOC to lead torich spin structures within the Mott state [24–28]. Butin these studies parameters were chosen to explore theimpact of Rashba SOC on the spin physics of Mott insu-lators while leaving the charge structure intact.In this work we explore Rashba SOC that is strongenough to cause the breakdown of charge ordering inMott insulators. This extreme limit is of direct relevanceto optical lattice experiments with synthetic SOC. Westudy, in particular, a 2D lattice model of two-componentinteracting bosons in the presence of tunable Rashba cou-pling. We find that strong Rashba SOC can cause thebreakdown of the Mott insulating state and drive a di-rect transition between the Mott insulator and a super-fluid state, even in the absence of single particle tunnel-ing between sites of the lattice [27]. This limit is thelattice version of the limit discussed above, h p i → h ~ ∇ V × ~p i ∼ E F , where vanishing kinetics leaves RashbaSOC to generate its own conducting state. For the caseof lattice bosons studied here, we find that Rashba SOCgenerates finite momentum superfluids. We show that α K F /E F h / E F GaAs InAsInSbAg(11), Au(111) Metallic SurfaceSrTiO /LaAlO Oxide Interface Cold Atoms
FIG. 1: Comparison of SOC strengths in solids and coldatoms. h , α , and E F denote the Zeeman energy, SOC co-efficient, and Fermi energy, respectively. For GaAs, the ef-fective mass is m ∗ = 0 . m [29], where m is the electronmass, the Rashba SOC strength is α = (0 . − . × − eV · m[30], and the g -factor is g ∗ = − . m ∗ = 0 . m [29], α = (0 . − . × − eV · m[32], and g ∗ = − . m ∗ = 0 . αK F = (1 . − . × − eV · m[34],and g ∗ = − m ∗ ∼ . m [2, 3], where the g -factor is assumed to be g ∗ = 2.The parameters for these four different examples are plot-ted at an external magnetic field of 5 Tesla. A high carrierdensity, n = 10 cm − , is used for the semiconductors. ForSrTiO /LaAlO oxide interfaces, the data are taken from Ref.35. Additional feasible parameter regimes are plotted as hor-izontal and vertical bars. these superfluids are characterized by staggered phasepatterns. We also find distinct superfluid states withstriped phase patterns that are separated by transitionson finite lattices with periodic boundaries. We predictthat finite momentum superfluids should be observablein time-of-flight measurements of the momentum distri-bution.The paper is organized as follows: In Sec. II we con-struct a Bose-Hubbard model of two-component atoms inthe presence of Rashba SOC. We also discuss two compli-mentary mean field approaches that allow us to computethe phase diagram, transition properties, and the mo-mentum distribution. In Sec. III we present results on fi-nite lattice sizes. We use Gutzwiller mean field theory toshow that Rashba SOC causes the Bosonic Mott insula-tor to give way to finite momentum superfluids. We alsoexplore inter-superfluid transitions. We find that transi-tions separate distinct phase patterns of finite momentumsuperfluids. We demonstrate in Sec. IV that these dif-ferent finite momentum phases can indeed be observedin experiments with a trapping potential. In Sec. Vwe present analytic arguments that transitions depend critically on boundary effects, akin to effects found inFulde-Ferrell-Larkin-Ovchinnikov (FFLO) superconduc-tors [36–48]. We show that analytic mean field calcula-tions in the infinite system size limit do not show thesetransitions. We summarize in Sec. VI. II. MODEL AND METHODS
We consider a 2D square optical lattice containingbosonic atoms with two hyperfine levels. States withtwo hyperfine levels act as a pseudo-spin 1/2 state. Wealso assume the presence of Raman beams that couplethe atomic momentum to the spin to generate syntheticSOC [7–11, 13]. The interaction between alkali atoms isgoverned by a short-range ( s -wave) repulsion. For a deepoptical lattice, the problem can be accurately describedin the single-band, tight-binding limit [49] where the s -wave interaction becomes an on-site Hubbard interactionand the SOC is discretized.To study this system we construct a Hubbard modelof two-component bosons in the presence of Rashba SOCon a square lattice. We allow the on-site Hubbard inter-action to have a spin-dependent interaction: H = − t X h ij i Ψ † i Ψ j + U X iσ n iσ ( n iσ − U ↑↓ X i n i ↑ n i ↓ − µ X iσ n iσ + iλ X h ij i Ψ † i ~e z · ( ~σ × ~d ij )Ψ j + H.c., (2)where, Ψ i = ( b i ↑ , b i ↓ ) T is a two-component bosonic an-nihilation operator at the site i , n iσ = b † iσ b iσ , t is thespin-independent nearest neighbor tunneling, U ( U ↑↓ ) isthe on-site interaction between bosons of the same (dif-ferent) spin σ , and µ is the chemical potential. In thelast term λ is the Rashba SOC strength, ~d ij is the unitvector between the neighboring sites i and j , and ~e z isthe unit vector along the z direction. In the following weuse U = 1 to set the energy scale.The tunneling and Rashba terms induce two differ-ent types of superfluidity. To see this we plot the spin-independent tunneling and spin-dependent tunneling inFig. 2. The left panel shows that the spin-independenttunneling favors phase uniformity since t is real. Butin the right panel we see that SOC has two effects: Itinduces tunneling between neighboring sites with twodifferent spin states and it imposes phase variation.The phase variation depends strongly on the directionof the neighboring sites. SOC therefore favors highlyanisotropic superfluid states. Without SOC the systemhas at least an U (1) ⊗ U (1) symmetry, which means thatthe total number of each species are conserved; however,SOC introduces spin flips between two neighboring sites,thus the system only respects U (1) symmetry and, asa result, the phase difference between the neighboring t tt t (a) (b) FIG. 2: Schematic of spin independent tunneling (a) and spin-dependent tunneling induced by SOC (b). In the later case,the tunneling takes place between two neighboring sites ac-companied by both spin flipping and phase variations. Thephase variation during tunneling is responsible for the cre-ation of the finite momentum superfluids. sites can not be gauged out. The competition betweenspin-independent tunneling and spin-dependent tunnel-ing tunes the transition between these different superflu-ids.In the weakly interacting limit the model exhibits threedifferent superfluid phases: In the regime when spin-independent tunneling dominates ( t ≫ λ ), the uniformsuperfluid is preferred and the total momentum of thesuperfluid is zero; In the opposite regime, a staggeredsuperfluid phase is preferred; and in the intermediateregime, t ∼ λ , the strong competition between the twotunnelings gives rise to superfluids with phase patternsthat depend strongly on boundary effects.Strong interactions add competing Mott insulatingphases and complicates estimates of the phase diagram.To study the competition between all ground states weuse two complimentary mean field approaches. We applythe Gutzwiller mean field method to finite system sizes(relevant to experiments) and compare with an otherwiseequivalent mean field method applied to infinite systemsizes.We now discuss the Gutzwiller mean field method[49, 50]. The method assumes a product ground stateof the form: | G i = Q i,σ (cid:16)P n f ( i,σ ) n | n i i,σ (cid:17) . This form forthe wavefunction has been extensively applied to bosonsin optical lattices [49], even in the presence of complexhopping amplitudes [51]. It generally gives quantitativelyreliable results in 2D and 3D, (for comparisons, see, e.g.,Ref. [52]), and is a particularly excellent approximationwhen computing local correlation functions (See, e.g.,Ref. [53]). The variational parameters f are obtainedby minimizing the total energy: E = h G | H | G ih G | G i . (3)We minimize the total ground state energy with the con-jugate gradient algorithm [54, 55]. The ground state en-ergy is reached when the energy variation is less than10 − U , which is sufficient to distinguish the energy dif-ference between different phases. (cid:1)(cid:1)(cid:2)(cid:3)(cid:4)(cid:4)(cid:2)(cid:3)(cid:5) ( (cid:6) ) ( (cid:7) ) (cid:1) (cid:1)(cid:2)(cid:1)(cid:3) (cid:1)(cid:2)(cid:4)(cid:1)(cid:1)(cid:2)(cid:3)(cid:4)(cid:4)(cid:2)(cid:3)(cid:5) ( (cid:8) ) (cid:1) (cid:1)(cid:2)(cid:1)(cid:3) (cid:1)(cid:2)(cid:4) ( (cid:9) ) FIG. 3: Phase diagrams of of Eq. 2 obtained from Gutzwillervariational simulations for an 8 × U ↑↓ = 0 , λ = 0, (b) U ↑↓ = 0 , λ =0 . U , (c) U ↑↓ = 0 . U, λ = 0, and (d) U ↑↓ = 0 . U, λ = 0 . U .The phase diagrams are determined by the amplitude of thespin-up superfluid order parameter. The spin-down superfluidorder parameter produces similar results. We supplement the finite system size Gutzwillermethod with an equivalent mean field limit applied toinfinite system sizes. We assume h b iσ i = ψe iθ iσ , where ψ is a real number. This assumption is equivalent to theassumed form for | G i but works best on infinite systemsizes. The total energy then becomes: E ψ = ( U + U ↑↓ ) ψ − ( U + 2 µ + tA + λB ) ψ , (4)where the coefficients are: A ≡ N − X h ij i h e i ( θ j ↑ − θ i ↑ ) + e i ( θ j ↓ − θ i ↓ ) + H.c. i , (5)and: B ≡ N − X h ij i h Z ∗ ij e i ( θ j ↓ − θ i ↑ ) − Z ij e i ( θ j ↑ − θ i ↓ ) + H.c. i , (6)with Z ij ≡ d xij + id yij and N is the number of sites. Animportant point here is that the total energy depends notonly on the magnitude of the order parameter ψ , but alsoon the phase difference between neighboring sites. We seethat the minimal energy E ψ corresponds to a maximalvalue of A and B when U , U ↑↓ , λ , and t assume positivevalues (the case studied in this paper). Here A dependsonly on the phase difference between the same spin states,while B depends strongly on the phase difference betweenspin up and spin down states in the neighboring sites.The competition between A and B governs competitionbetween superfluids with distinct phase patterns. When λ = 0, A takes its maximum value when all of the siteshave the same phase, which corresponds to the uniformsuperfluid phase. III. QUANTUM PHASES IN FINITE LATTICESWITH PERIODIC BOUNDARIES
We now discuss results that demonstrate the competi-tion between various Mott and superfluid phases in thepresence of SOC. We first present our results on small sys-tem sizes with periodic boundaries. These system sizesare consistent with small states formed in the center oftraps in experiments.Fig. 3 shows the phase diagram for four different lim-its of the model, Eq. (2). Fig. 3a plots the Bose-Hubbard phase diagram [56] that results from setting theSOC term and the inter-spin interaction term to zero inEq. (2), i.e., λ = U ↑↓ = 0. The absence of inter-spin inter-actions allows two identical copies of the Mott insulator.The lower and upper Mott lobes in Fig. 3a correspond to h n i ↑ i = h n i ↓ i = 1 and h n i ↑ i = h n i ↓ i = 2, respectively.Fig. 3c shows the result of adding inter-spin repulsion, U ↑↓ >
0, but with no SOC, λ = 0. Here we see that thatthe original low energy Mott lobe is pushed up. The ap-pearance of the small Mott lobes (above and below thelarger Mott lobe) correspond to the formation of Mottinsulators with Ising-type spin ordering. To see this, werewrite the interaction terms in H using sum and differ-ence operators, n i ± ≡ n i ↑ ± n i ↓ . The large Mott lobe inFig. 3c then corresponds to h n i + i = 2 , h n i − i = 0. Theupper and lower small Mott lobes exhibit degeneracies(for t = 0) and correspond to h n i + i = 3 , h n i − i = ± h n i + i = 1 , h n i − i = ±
1, respectively. Here we excludesuper exchange effects, O ( t /U ), discussed in other work[24–26, 28].We now discuss the phase diagram that results fromadding SOC. Figs. 3b and 3d plot the phase diagramsthat result from adding SOC to the states depicted inFigs. 3a and 3c, respectively. In both figures we seethat the Mott insulators at higher µ vanish. Increas-ing µ causes a direct transition from a Mott insulator toa SOC-generated superfluid. At t = 0, SOC alone drivesthe formation of a superfluid. We find that the Mottinsulators that normally persist at t = 0 for all µ areactually supplanted by SOC-generated superfluids. The t = 0 superfluids found on this part of the phase diagramderive kinetics purely from the spin-dependent tunnel-ing in SOC. We therefore find that even in the limit ofvanishing kinetics, the Rashba effect drives the Mott in-sulator into a conducting state (in this case, a superfluidstate). We have also checked the phase diagrams of 4 × × × λ and t , as well as boundary effects.Fig. 4 shows the transitions of different superfluidphase patterns. The left column shows the order param-eters for the 8 × t ≤ . U . For t > . U ,the order parameter gradually increases with t , which FIG. 4: Plot of the amplitude of spin-up superfluid orderparameter |h b ↑ i| , the filling factor h n ↑ i and the energy density e as a function of the spin-independent tunneling at U ↑↓ =0 . U , λ = 0 . U and µ = 1 . U for periodic (left panel) andopen (right panel) boundary conditions. (cid:1) (cid:1) - (cid:1) - (cid:1) (cid:1) (cid:1) ( (cid:1) ) - (cid:1) (cid:1) (cid:1) ( (cid:2) ) - (cid:1) (cid:1) (cid:1) ( (cid:3) ) FIG. 5: (Color online) Spin-dependent momentum distri-bution, Eq. (7), for different superfluids at U ↑↓ = 0 . U , λ = 0 . U , µ = 1 . U , (a) t = 0 . U , and (b) t = 0 . U . indicates a transition between different superfluids at t ∼ . U . For the open boundary condition case shownin the right column, there is no such transition since thephase can vary smoothly over the lattice.The superfluids with different phase patterns havedifferent momenta. To see this we compute the spin-dependent momentum distribution at wavevector k : h ρ ↑ , ↓ ( ~k ) i = N − X i,j h b † i ↑ b j ↓ i e i~k · ( ~R i − ~R j ) , (7)where the lattice spacing is chosen as the unit of distanceand ~R j is the location of the lattice site j .We take random initial guess states and minimize thetotal energy to compute the ground state | G i , with whichthe spin-dependent momentum distribution is computedas h G | ρ ↑ , ↓ ( ~k ) | G i / h G | G i . We get four degenerate groundstates with different momentum distributions, where the D symmetry of the lattice system is spontaneously bro-ken. Similar results have been discovered in the contin-uum model of spin-1 / λ and t . In the non-interacting limit, the ground state energyof the system with SOC is E = − t (cos k x + cos k y ) − λ q sin k x + sin k y . The energy minima are located at k = (cid:0) ± arctan( λ/ √ t ) , ± arctan( λ/ √ t ) (cid:1) . On a finite8 × k can only take discrete values. In Par-ticular, for λ/t = 0 .
8, the energy minima are located at(0 , π/ , − π/ π/ ,
0) and ( − π/ , . In the pres-ence of interactions, D symmetry is spontaneously bro-ken and the system chooses one of the minima in Fig.5(b). Similarly, for λ/t = 8, the the energy minima are k = ( ± π/ , ± π/ IV. QUANTUM PHASES IN A TRAPPINGPOTENTIAL
We now consider the effects of realistic confinementon the superfluid transitions. The finite momentum su-perfluids considered here are akin to the FFLO phasediscussed in the context of trapped atomic Fermi gases.The FFLO state depends strongly on lattice geometry.Finite size effects are normally not considered to be rel-evant in solids because system sizes are typically muchlarger than correlation lengths. But cold atomic gasescan be put into regimes where the system size is on theorder of superfluid correlation lengths.Small magneto-optical trapping potentials can be cre-ated in cold atom systems. We add a spatially varyingchemical potential term to Eq. (2) to model confinement: P i V ( ~R i )( n i, ↑ + n i, ↓ ). The trapping potentials are wellapproximated by a parabolic potential. We consider: V ( ~R i ) = 0 . U "(cid:18) R xi − L x − (cid:19) + (cid:18) R yi − L y − (cid:19) (8)where R xi ( R yi ) is the x ( y ) coordinate of site i and L x ( L y )is the lattice size along the x ( y ) direction. The trap coef-ficient is chosen to ensure that the trapped atom densityvanishes before the edge of the lattice is reached. Within (cid:1)(cid:2)(cid:1)(cid:3)(cid:1)(cid:4)(cid:1) ( (cid:5) ) ( (cid:6) ) ( (cid:7) ) (cid:1) (cid:1) - (cid:2) - (cid:2) (cid:1) (cid:2) (cid:1) (cid:1) (cid:2) (cid:1)(cid:2)(cid:1)(cid:3)(cid:1)(cid:4)(cid:1) ( (cid:8) ) ( (cid:9) ) ( (cid:10) ) |< (cid:2) (cid:1) >| (cid:2)(cid:11)(cid:1)(cid:1)(cid:11)(cid:14)(cid:1)(cid:11)(cid:13)(cid:1)(cid:11)(cid:12)(cid:1)(cid:11)(cid:3)(cid:1)(cid:1) (cid:2)(cid:1) (cid:3)(cid:1) (cid:4)(cid:1)(cid:2)(cid:1)(cid:3)(cid:1)(cid:4)(cid:1) ( (cid:15) ) ( (cid:16) ) ( (cid:17) ) (cid:1) < (cid:3) (cid:1) > (cid:1)(cid:1)(cid:11)(cid:18)(cid:2)(cid:11)(cid:1)(cid:1) (cid:3) (cid:2)(cid:1) (cid:3) (cid:3)(cid:1) (cid:3) (cid:4)(cid:1) (cid:3) (cid:1) (cid:3) (cid:2)(cid:1) (cid:3) (cid:3)(cid:1) (cid:3) (cid:4)(cid:1) (cid:3) FIG. 6: Correlation functions of finite momentum superfluidson a 32 ×
32 lattice with a confining potential [Eq. 8] for µ = 0 . U , U ↑↓ = 0 and λ = 0 . U . The left column showsresults for t = 0 . U , the middle column for t = 0 . U andthe right column for t = 0 . U . The top three panels plotthe phase φ ↑ of the spin up superfluid order parameter. Themiddle three panels plot the magnitude and the bottom threepanels plot the density. The phase patterns in the top twopanels reveal a sudden change in superfluid order. the mean-feild theory, we can compute the local superluidorder parameter in the trap h b i,σ i = P n √ nf ( i,σ ) ∗ n − f ( i,σ ) n . The local density is obtainted as h n i,σ i = P n n | f ( i,σ ) n | . We now show that the phase change, discussed in pe-riodic systems above, also manifests in trapped systems.Fig. 6 shows a typical example obtained from solvingEq. (2) in the presence of parabolic trapping using theGutzwiller ansatz with 10 random initial guess states.Since Mott insulator is a incoherent sate with randompahses, phases of uparrow superfluid order parameterwith |h b ↑ i| ≤ .
05 are plotted with dark grey color inthe top panel of Fig. 6. As the hopping parameter in-creases, the phase reorients in the trap from non-uniformpattern to uniform due to the SOC effect. The effects pre-dicted here are observable in measurements sensitive tothe phase of the superfluid order parameter (e.g., the mo-mentum distribution function). This calculation showsthat realistic trapping potentials lead to finite sized sys-tems that harbor the transitions found in periodic sys-tems discussed above.
V. QUANTUM PHASES IN INFINITELATTICES
So far our study has been limited to finite-sized lat-tices. Here boundary effects put a strong constraint E k / U Infinity 6 x 65 x 54 x 4
FIG. 7: Plot of the kinetic energy terms, Eq. (9), as a functionof the spin-independent tunneling at λ = 0 . U , for a 4 × ×
5, 6 × on the superfluid phase patterns that can be realized.But we can use Eq. (4) to study infinite lattices. Wefind a general solution for the lowest-energy state, θ i ↑ = α ( R yi − R xi ) and θ i ↓ = π + α ( R yi − R xi ), where α =arctan( λ √ / t ). The corresponding energy for just thekinetic terms is: E k = − ( tA + λB ) ψ (9)The competing superfluids arise from the competition be-tween A and B coefficients.Before studying the infinite system case we first testthat Eq. (9) gives the same results as the Gutzwiller meanfield theory. We find that this is the case by comparingresults obtained from maximizing tA + λB in Eq. (9) ona finite lattice with the Gutzwiller mean field theory. Wefind precisely the same phase patterns given in Fig. 5.This confirms that the Gutzwiller mean field theory isequivalent to Eq. (9) on finite lattices.We now study infinite lattice sizes. In the infinite sys-tem size limit we find: E k → − √ λ sin( α ) − t cos( α ).This implies that the energy will change smoothly as theperiod of the finite momentum superfluids changes dra-matically. Fig. 7 shows that the energy computed onthe infinite system size is in fact smooth. We thereforeconclude that infinite lattice sizes will eliminate transi- tions observed in finite sized systems. A similar resultwas found in studies of FFLO superfluids where peri-odic boundaries also constrain the FFLO momentum toselect certain values [60, 61]. But we note that realis-tic experiments are actually trapped finite sized systemswith N ∼ − . We therefore conclude that transi-tions between distinct superfluids found here should beobservable in the small system limit defined by the trapcenter. VI. SUMMARY
We have studied the interplay of strong interactionand Rashba SOC in a model motivated by optical lat-tice experiments: a 2D Hubbard model of two-componentbosons. We used mean field theory to map out thephase diagram and study transitions. We find that strongRashba SOC can completely destroy the Mott insulatorstate, even in the absence of spin-independent tunnel-ing in the lattice. The Rashba SOC leads to superflu-ids with complex phase patterns and finite momentum.We identified transitions between superfluids with twodifferent staggered phase patterns, that can be identi-fied in the spin-dependent momentum distribution. Thespin-dependent momentum could be accessed in time-of-flight measurements on optical lattices. We expect thesetransitions to occur in finite sized systems but the phasepatterns and precise momenta depend strongly on theboundaries. We checked that these transitions in phasepatterns become smooth in infinite system sizes.Our work relates to the nature of Mott insulator statesin solids. Our study of a 2D lattice finds that it is in prin-ciple possible for strong Rashba SOC to convert a Mottinsulator into a conducting state even in the limit of van-ishing kinetics ( t → λ ∼ h p i → h ~ ∇ V × ~p i ∼ E F in the continuum). Thislimit could have bearing on the nature of 3D Mott in-sulator surface states that experience very weak kineticsbut strong electric fields. Acknowledgements
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