Magnetic supersolid phases of two-dimensional extended Bose-Hubbard model with spin-orbit coupling

The study of ultracold atomic spin systems with long-range interaction provides the possibility of searching for magnetic supersolid phases in quantum many-body scenarios. In this paper, we consider two-species Bose gases with spin-orbit coupling and nearest-neighbor interaction confined in a two-dimensional optical lattice. The competition between spin-orbit coupling and interactions creates rich ground-state diagrams with supersolid phases exhibiting phase modulations or magnetic orderings. We obtain the phase-twisted and phase-striped pair checkboard supersolid phases that are generated by the combination of spin-orbit coupling and intraspecies nearest-neighbor interaction. The introduction of interspecies nearest-neighbor interaction enriches the quantum phases of the system. It leads to the appearance of the phase-twisted and phase-striped lattice supersolid phases. In addition to the lattice supersolid phase, we find the emergence of nontrivial supersolid phases that depend on the interspecies on-site interaction strength. The lattice-insulated supersolid phase with supersolidility in one species but insulation in the other exists in the miscible domain, while the pair striped supersolid phase with stripe structures in each species is in the immiscible domain. Finally, to further characterize each phase, we discuss their spin-dependent momentum distributions and spin textures. The magnetic textures, such as antiferromagnetic, spiral and stripe orders, are shown in SS phases. The results here could help in the observe for these magnetic supersolid phases in ultracold atomic experiments with nearest-neighbor interaction and spin-orbit coupling in optical lattice.

Ultracold atoms with spin-orbit coupling (SOC) represent an important and active research field in quantum gas physics.Recently, the artificial SOC effect in multispecies Bose systems has been realized in the cold atomic experiments by tuning the Raman field [40][41][42].The form of SOC can be of either the Rashba [43] or Dresselhaus [44] type, both of which are frequently analyzed in terms of an effective gauge force.The combination of SOC and the interaction of atoms gives rise to a variety of quantum states.The effective super-exchange spin model with the Dzyaloshinskii-Moriya type (DM-type)interactions can be obtained by the second-order perturbation theory [45,46] in the MI regime of two-dimensional (2D) spin-orbit coupled Bose-Hubbard model.The spiral, vortex crystal and skyrmion crystal magnetic structures are found by applying the classical Monte-Carlo (MC) simulations, bosonic dynamical mean-field (BDMF) theory, variational order (VO) method and tensor network states (TNS) method [47][48][49][50][51][52][53][54][55][56][57].The effects of the strength and symmetry of SOC on the SF phase and MI-SF phase transition are also investigated.The phase-twisted SF (PT-SF) phase, phase-striped SF (PS-SF) phase, orbitalordered SF phase and striped SS phase are driven by SOC [58][59][60][61][62][63][64][65][66].However, the comprehensive theoretical study of ground-state phase diagrams and phase transitions in a 2D spin-orbit coupled Bose-Hubbard model with NN interaction is still missing.
In this work, we investigate the quantum phases and phase transitions of 2D extended Bose-Hubbard model with SOC by using the inhomogeneous dynamical Guztwiller mean-field (IDGMF) method.The competition between SOC and interactions (including on-site and NN interactions) gives rise to a variety of quantum phases with phase modulation or spin ordering.The translational symmetries of each species density are broken by the intraspecies NN interaction.The pair checkboard SS (PCSS) phase with checkboard structure in each species and uniformly in total density appeared when only considering intraspecies NN interaction.The SOC drives the phase-twisted PCSS (PT-PCSS) and phase-striped PCSS (PS-PCSS) phases.The introduction of interspecies NN interaction enriches the quantum phases of the system.The phase-twisted lattice SS (PT-LSS) and phase-striped lattice SS (PS-LSS) phases are preferred.For the lattice SS (LSS) phase, the translational symmetries of both each species and total densities are broken by the interspecies NN interaction, and the lattice structure stably exists in total density.We find that the interspecies on-site interaction plays a dominant role in the quantum phases and phase transitions.The latticeinsulated SS (LISS) phase with supersolidility in one spin species but insulation in the other exists in the miscible domain (U 2 ↑↓ < U ↑↑ U ↓↓ ) [67], while pair striped SS (PSSS) phase with stripe structure in the immiscible domain (U 2 ↑↓ > U ↑↑ U ↓↓ ) [68].Unlike the PCSS phase, the PSSS phase is characterized by the stripe structure of density of each species.The SOC also drives the phasetwisted PSSS (PT-PSSS) and phase-striped (PS-PSSS) phases.Therefore, there is a transition from the LSS phase to the phase-twisted SF (PT-SF) phase, and to the phase-striped SF (PS-SF) phase in the miscible domain, and from the LSS phase to the PT-PSSS phase, and to the PS-PSSS phase in the immiscible domain.Finally, to further characterize each phase, we have discussed their spin-dependent momentum distributions and spin textures.The magnetic textures, such as antiferromagnetic (AFM), spiral and stripe orders, are shown in the SS phases.The results here could help in the observe for these magnetic SS phases in ultracold atomic experiments with NN interaction and SOC in optical lattice.
The paper is organized as follows: In Sec.II, we introduce the model of the spin-orbit-coupled two-species Bose gases in a 2D optical lattice with NN interaction.In Sec.III, we display MI-SF phase transition of spin-orbit coupled Bose-Hubbard model.In Sec.IV, the phase diagrams and phase transitions of 2D spin-orbit coupled extended Bose-Hubbard model without and with interspecies NN interaction are discussed in sections A and B, respectively.A summary is included in Sec.V.

II. MODEL AND HAMILTON
To study the quantum phases and phase transitions of this system, we construct a two-component Bose-Hubbard model in the presence of SOC and NN interaction on a 2D square lattice.In the tight-binding form, the Hamiltonian can be written as where σ =↑, ↓ denotes the spin-σ species and (p, q) are the sites indices.b †σ p,q ( bσ p,q ) is bosonic creation (annihilation) operator, nσ p,q is the bosonic number operator and µ σ p,q is the chemical potential of spin-σ species at site (p, q).t x (t y ) and γ x (γ y ) are the hopping strength and SOC strength along the x (y) direction, respectively.U σσ and V σσ is the intraspecies on-site and NN interactions of spin-σ species, respectively.For simplicity, we choose symmetric hopping t x = t y = t and SOC γ x = γ y = γ, identical intraspecies on-site (NN) interaction ) and equal chemical potential µ ↑ p,q = µ ↓ p,q = µ.U ↑↓ and V ↑↓ are the interspecies on-site and NN interactions, respectively.
The bosonic operators can be transformed by Fourier bσ k e ikj , which satisfy the commutation relations [ bσ k , bσ k ′ ] = δ kk ′ .In the limit of U ≪ t and V ≪ t, the Hamiltonian of Eq. (1) in the momentum space is where H k = −2t(cos k x + cos k y ) Î + 2γ(sin k y σx − sin k x σy ).The energy eigenvalues of H k are The four degenerate minima in the lower branch are ±Q = (±k 0 , ±k 0 ) with k 0 = arctan γ √ 2t .The corresponding eigenstates are Obviously, the location of the minima of Bose gases determined by SOC, which shows the SOC effect plays an important role on the ground-state phases of spin-orbit coupled Bose system.The ground-state phases and phase transitions of the extended Bose-Hubbard model with SOC in Eq. ( 1) can be obtained by using the IDGMF method.Under the mean-field decoupling approximation, the hopping and NN interaction terms can be written as b †σ The many-body wave function of the ground state of the system is given by where |ψ p,q is the single site ground-state.|n ↑ , n ↓ p,q is the Fock state and c is the the probability amplitudes, which is normalized in our numerical simulations, i.e., The truncation of maximum number of bosons at each lattice site n max = 6 in the numerical simulation.The SF order parameters of spin-σ species at site (p, q) are obtained as by using the above ansatz and the filling numbers are The c n ↑ ,n ↓ p,q is complex with SOC, therefore, the SF order parameters are complex numbers in general.It can be rewritten in terms of the magnitude and phase, i.e., ∆ σ p,q = |∆ σ p,q |e iθ σ p,q .Since U ↑↑ = U ↓↓ and µ ↑ p,q = µ ↓ p,q , the SF order parameters

Minimization of the effective action Ψ|i
NN sites of site (p, q).The system size Ω = L × L lattice sites with the periodic boundary conditions, here, we choose L = 12.The ground-state phases and phase transitions are obtained by using the standard imaginarytime-evolution propagation [74][75][76] in Eq. ( 9), i.e., t → −it.In order to have universality, we choose the random number as the initial Guztwiller wave function.

III. MI-SF PHASE TRANSITIONS IN SPIN-ORBIT COUPLED BOSE-HUBBARD MODEL
We first discuss the effect of SOC γ and on-site interspecies interaction U ↑↓ on the ground-state phases and phase transitions in the standard spin-orbit coupled Bose-Hubbard model, i.e., V = 0 and V ↑↓ = 0. Figure 1 shows the phase diagrams in the t/U − µ/U plane for different values of γ with U ↑,↓ /U = 0.8 in (a) and U ↑,↓ /U = 1.2 in (b).The MI phases are characterized by MI(N ), where N = n ↑ + n ↓ ∈ N. Two quantum phases: MI and SF phases exhibited in the absence of SOC γ = 0, which are similar to the single-species Bose-Hubbard model [77].The lobe sizes of MI(N ∈ 2n + 1) are smaller than of those MI(N ∈ 2n) at U ↑,↓ < U , while the lobe sizes of MI(N ) shrink as N increases at U ↑,↓ > U , as shown in Figs.1(a1) and 1(b1), respectively.Though the magnitudes of SF order parameters are uniform at each site, the phases θ p,q =arg(∆ p,q ) are nonuniform due to SOC.The PT-SF phase that phase varies diagonally across the sites and the PS-SF phase that phase exhibits stripelike patterns along the axis direction appeared in the presence of SOC, as shown in Figs.4(a) and 4(b).They also can be classified by using the spin-dependent mo- [65,66], where r A (r B ) is the location of A-th (B-th) site, site A and site B are the NN sites.The spin-dependent momentum peak at ρ ↑↓ (−k 0 , −k 0 ) or ρ ↑↓ (k 0 , k 0 ) along the diagonal direction in the PT-SF phase and ρ ↑↓ (k 0 , 0) or ρ ↑↓ (−k 0 , 0) ( ρ ↑↓ (0, k 0 ) or ρ ↑↓ (0, −k 0 ) ) along the x (y) direction in the PS-SF phase.The phenomena illustrate that only one of the four states of Q is occupied, the PT-SF phase chooses the position of the diagonal of the Brillouin zone and the PS-SF phase chooses the axis direction of k-space.The SOC shrinks the MI lobe size and it vanishes as the SOC strength is increased beyond a critical value γ c .The phase transitions from the PT-SF phase to the PS-SF phase, and to the zero momentum SF (ZM-SF) phase at U ↑↓ /U = 0.8, and to the z-polarized ferromagnetic SF (Z-SF) phase at U ↑↓ /U = 1.2 with the hopping strength increases.
The critical hopping t c of MI-SF transition in spinorbit coupled Bose-Hubbard model can be given by the second-order perturbation theory (details given in Appendix A), where t 0 = t ↑ 0 = t ↓ 0 is the critical hopping of MI-SF transition without SOC, and z = 4 is the NN site number.For the MI(N ∈ 2n) phase, the occupy number n ↑ = n ↓ , [37,78].For the MI(N ∈ 2n + 1) phase, one atom at each site is chosen randomly from the two species, and the energies of the system degenerate for all the possible combinations.The occupy number (n The phase boundaries (filled red circle lines) of MI-SF phases calculated by solving Eq. ( 10) are agreement with the numerical simulation results in Fig. 1.
The magnetic structures of the PT-SF and PS-SF phases are also studied in Fig. 8(a) and (b), respectively.The spin texture is defined by [79] and σζ is the Pauli matrix.The PT-SF and PS-SF phases show the interesting spin configurations.The spiral order is exhibited in the PT-SF phase and stripe order in the PS-SF phase in Fig. 8(a) and (b), respectively.The spiral order is the spins having a spiral wave along the diagonal direction and the stripe order is the spins being separated by periodically spaced domain walls along the axis direction.The spin texture structures are the same as the SF order phase distributions in Figs.4(a) and 4(b).The values of S z are weak, i.e., S z ∈ {−0.12, 0.12} in the PT-SF phase and {−0.34, 0.04} in the PS-SF phase.

IV. MAGNETIC SS PHASES IN EXTENDED BOSE-HUBBARD MODEL WITH SOC
The density translational symmetry of the system can be spontaneously broken by the long-range NN interaction, and the quantum phases with periodic density modulations emerge, such as the DW and SS phases.The SOC can induce the exotic magnetic orders and SF order phase structures.Therefore, we study the quantum phases and phase transitions of extended Bose-Hubbard model with SOC.
We first discuss the ground-state phase of spin-orbit coupled Bose-Hubbard model with intraspecies NN interaction, i.e., V ↑↓ = 0. We plot the phase diagrams as functions of t and µ for different V and γ at U ↑↓ /U = 0.8 in Fig. 2

DW phase ∆n
. The translational symmetry of single species is broken by intraspecies NN interaction V ↑↓ .As a result, a type of SS phase with checkerboard structure in single species appears, and hence can be regarded as PCSS phase.The checkerboard structure of each species makes the intraspecies NN interaction less influential on the ground-state phases, for example, (n ↑ A , n ↓ A ) = (0.1, 1.1) and (n ↑ B , n ↓ B,q ) = (1.1,0.1) in Fig. 4(c), the intraspecies NN interaction term in Eq. ( 1) is weak that can not enough to break the translational symmetry of total density, therefore, the total density N = n ↑ + n ↓ of PCSS phase is uniform.When γ = 0, the DW(1,0), MI(1,1), DW(2,1) phase appear at U ↑↓ /U = 0.8 and V 1 /U = 0.05 in Fig. 2

(a1). If the interspecies on-site interaction or intraspecies NN interaction is increased beyond a critical value
1 , only DW(N ∈ N,0) phase exists.The domain of the PCSS phase also increases, one can be seen in Figs.2(b1) and 3(a1).SOC-driven the PT-PCSS and PS-PCSS phases in Figs.4(c)-(e).In addition to the PT-PCSS and PS-PCSS phases, the PT-SF phase or PS-SF phase is also observed for weak intraspecies NN interaction V 1 /U < ∼ 0.12.Upon increasing V 1 further, the phase variations of SF order are inhibited by intraspecies NN interaction, the zero momentum PCSS (ZM-PCSS) phase (see Fig. 4(e)) with θ σ p,q = 0 occupies the most region, as shown in Figs.2(c)-(d

) and 3(c)-(d).
The magnetic structures of PT-SCSS and PS-SCSS phases are shown in Figs.8(c)-(d), respectively.The value of S z ∈ {−1, 1}.The PT-PCSS phase shows the AFM order along the z-axis (Z-AFM) that the neighboring spins point to the opposite directions (S z = ±1).The PS-PCSS phase also shows the antiferromagnet order structure, however, the vectors form a certain angle to the z-axis due to the competition of hopping and SOC, one can be seen in Fig. 8(d).
The effect of intraspecies NN interaction V 1 and SOC γ on the ground-state phases has been discussed above.Two kinds of the PCSS phases, i.e., the PT-PCSS and PS-PCSS phases with periodic density modulation in each species are found.Here, we study the quantum phases of spin-orbit coupled Bose-Hubbard model by adding the interspecies NN interaction.For simplicity, we consider symmetric NN interactions, i.e., The ground-state phase diagrams in the t − µ plane for different V and γ are shown in Fig. 5 with U ↑↓ /U = 0.8 and Fig. 6 with U ↑↓ /U = 1.2.The DW and MI phases appear alternately with µ increasing without SOC γ/U = 0 at weak V /U = 0.05, as shown in Figs.5(a1) and 6(a1).The DW lobes are surrounded by a thin envelope of a new kind of SS phase.The SS phase has the pe-riodic density modulations in both each species and total densities.The total density exhibits the lattice structure, we take it as the LSS phase.SOC-driven the PT-LSS and PS-LSS phases, one can be seen in Figs.7(d) and 7(e).Two peaks of spin-dependent momentum at ρ ↑↓ (k 0 , k 0 ) and ρ ↑↓ (−k 0 , −k 0 ) with equal heights along the diagonal direction in the PT-LSS phase and ρ ↑↓ (k 0 , 0) and ρ ↑↓ (−k 0 , 0) ( ρ ↑↓ (0, k 0 ) and ρ ↑↓ (0, −k 0 ) ) along the x (y) direction in the PS-LSS phase.An interesting phenomenon is shown in the regime around the DW(3,2) in Fig. 5(a1), the LISS phase with supersolidity in one spin species but insulation in the other appears.The density and spin-dependent momentum of the PT-LSS phase are shown in Fig. 7(a).The SOC also shrinks the DW and MI lobes, and only the MI phase survives at γ/U = 0.04.The reason for the existence of the MI phase at larger SOC is that the energy consumption of the MI phase is larger than the DW phase due to repulsion between two species coexisting on the same lattice site at finite U ↑↓ .We find that the appearance of some ground-state phases depending on the interspecies on-site interaction of spin-orbit coupled extended Bose-Hubbard model at larger hopping strength t.The PT-SF and PS-SF phases emerge in the immiscible domain in Fig. 5 with U ↑↓ /U = 0.8 while the PT-PSSS and PS-PSSS phases in the immiscible domain in Fig. 6 with U ↑↓ /U = 1.2.For the PT-PSSS or PS-PSSS phase, each species occupies opposite wave vectors of the four states of Q, the stripe structures in single species density and uniform in total density.Two peaks of the spin-dependent momentum are exhibited in PT-PSSS and PS-PSSS phases, one can be seen in Figs.7(b) and 7(c).For larger V , the PT-SF and PS-SF (PT-PSSS and PS-PSSS) phases are replaced by the PT-LSS, PS-LSS and zero-momentum LSS (ZM-LSS) phases.Similar to the case of in section A only with intraspecies NN interaction, the intraspecies and interspecies NN interactions also inhibit the phase variation of SF order of LSS phase, ZM-LSS phase (see Fig. 7(f)) occupies the most region, as shown in Figs.

5(d) and 6(d).
The spin textures of the PT-PSSS, PS-PSSS, PT-LSS, and PS-LSS phases are respectively shown in Figs.8(e)-(h).The PT-SS phase favors the spiral order and the PS-SS phase is the stripe order.The combination of the NN interactions and SOC plays an important role on the spatial period of spiral orders.The spiral order of the PT-PSSS phase has spatial periods 10 sites while the PT-LSS phase has 5 sites, which can be respectively denoted as spiral-10 and spiral-5 orders, as shown in Figs.8(e) and 8(g).

V. SUMMARY
We have investigated the quantum phases and phase transitions of spin-orbit coupled Bose gases in a 2D extended Bose-Hubbard model by using IDGMF method.The competition between SOC and interactions creates rich ground-state diagrams with SS phases exhibiting phase modulations or magnetic orderings.The combined effect of intraspecies NN interaction and SOC results in the PT-PCSS and PS-PCSS phases.The PCSS phase only has the periodic density modulation in each species and is uniform in total density.The introduction of interspecies NN interaction enriches the quantum phases of the system.The PT-LSS and PS-LSS phases with periodic density modulation in both each species and total densities are preferred.We find that the appearance of some ground-state phases depend on interspecies onsite interaction.The LISS phase with supersolidity in one spin species but insulation in the other exists in the miscible domain, while the PSSS phase with stripe structures in each spin species in the immiscible domain.For the PT-or PS-PSSS phase, each species occupies opposite wave vectors of the four states of the single-particle energy spectrum, it shows the stripe structures in each species density and uniform in total density.Finally, to further characterize each phase, we discuss their spindependent momentum distributions and spin textures.The magnetic textures such as AFM, spiral and stripe orders are shown in these SS phases.The spiral orders also can be classified by the spatial periods, including the spiral-10 and spiral-5 orders.The results here could help in the observe for these magnetic SS phases in ultracold atomic experiments with NN interactions and SOC in optical lattice.We first discuss the spin-orbit coupled Bose-Hubbard model, the hopping and SOC terms in the single-site Hamiltonian regarded as the perturbation Hamiltonian and the on-site interaction terms with the chemical potential as the unperturbed Hamiltonian.Therefore, the energy of the ground state of the unperturbed Hamiltonian is given as The second-order perturbed ground-state energy can be written as where Φ a † = (∆ ↑ † p,q , ∆ ↓ † p,q ).The perturbation Hamiltonian T a p,q = −t σ ∆σ p,q ( b †σ p,q + bσ p,q ) − |∆ σ p,q | 2 + γ ∆↑ p ′ ,q ( b †↓ p,q + b↓ p,q ) − ∆↓ p ′ ,q ( b †↑ p,q + b↑ p,q ) + iγ ∆↑ p,q ′ ( b †↓ p,q − b↓ p,q ) + ∆↓ p,q ′ ( b †↑ p,q − b↑ p,q ) , (a3) where ∆σ p,q = ∆ σ p−1,q +∆ σ p+1,q +∆ σ p,q−1 +∆ σ p,q+1 = z∆ σ p,q , ∆σ p ′ ,q = ∆ σ p−1,q +∆ σ p+1,q and ∆σ p,q ′ = ∆ σ p,q−1 +∆ σ p,q+1 .λ 1 and λ 2 are the eigenvalues of matrix A. The parameter , where t σ 0 is the critical hopping of MI-SF transition in the absence of SOC of spin-σ species.For the MI phase n ↑ = n ↓ , the boundaries t ↑ 0 = t ↓ 0 .If we want to obtain the ground-sate phases, we should min{E n ↑ p,q ,n ↓ p,q ∂∆ ↑ p,q = 0 and ∂E n ↑ p,q ,n ↓ p,q ∂∆ ↓ p,q = 0. Therefore, the eigenvalues λ 1 = λ 2 = 0.
where D = z 2 t 2 +2γ 2 zt , thus, µ 2 + U − 2U n ↑ p,q − 2U ↑↓ n ↓ p,q + D µ + U n ↑2 p,q − U 2 n ↑ p,q + U U ↑↓ n ↑ p,q n ↓ p,q + U ↑↓ n ↓2 p,q + D(U − U ↑↓ n ↓ p,q ) = 0. (a6) We obtain Here µ ↑ p,q = µ ↓ p,q .The critical condition for the MI-SF transition of each species is when the terms under the square root in Eq. (a5) vanish or when µ σ p,q− = µ σ p,q+ .We yield the critical values of the spin-orbit coupled Bose-Hubbard model as B: Spin-orbit coupled Bose-Hubbard model with NN interactions For the extended Bose-Hubbard model with SOC, the hopping and SOC terms in the single-site Hamiltonian are also the perturbation Hamiltonian, and the interactions (the on-site and NN interactions) with the chemical potential are the unperturbed Hamiltonian.The energy of the ground state of the unperturbed Hamiltonian is given as
and U ↑↓ /U = 1.2 in Fig. 3. Here, the DW and MI phases can be described by the NN lattice sites occupation number (n ↑ A , n ↑ B ).They have zero superfluid order parameter |∆ σ A | = |∆ σ B | = 0, hence are incompressible.The MI phase has an integer commensurate occupation number n σ A = n σ B ∈ N while the DW phase with n σ A = n σ B .The relative occupation number of the
MI-SS or DW-SS phase transition in two-species extended Bose-Hubbard model of spin-σ species at sites A and B, respectively.When γ = 0, the Eq. (
This work is supported by the Scientific and Technological Research Program of the Education Department of Hubei province under Grant Nos.D20222502, the NSF of Hubei Province of China under Grant No. 2022CFB499, the NSF of China under Grant No. 11904242 and the Talent project of Hubei Normal University under Grant No. HS2022RC033.APPENDIX: PERTURBATIVE TREATMENT A: Spin-orbit coupled Bose-Hubbard model

FIG. 8 :
FIG. 8: (Color online) The spin textures of the PT-SF phase, PS-SF phase, PT-SCSS phase, PS-SCSS phase, PT-STSS phase, PS-STSS phase, PT-CSS phase and PS-CSS phase are respectively shown in (a)-(h).The (Sx, Sy) speciess have been plotted using arrows, while the Sz species has been plotted in color.