Variational study of phase diagrams of spin–orbit coupled bosons

We use a variational order method to explore the ground state phase diagrams of spin–orbit coupled Bose atoms in optical lattices. By properly parameterizing the order with an ansatz function for each possible phase and comparing their energies which are minimized with respect to the variational parameters, we can effectively identify the lowest energy states and depict the phase diagram. While the Mott-insulator-superfluid phase boundary is computed by the usual Gutzwiller method, the spin-ordered phase structures of the deep Mott regime as well as the superfluid regime are studied by the variational order method. For the isotropic spin–orbit coupling, the phase diagram in the deep Mott insulator regime is qualitatively similar to results derived by the classical Monte-Carlo simulations or other numerical methods. The spiral phases with different spatial periods are discussed in detail. We find three new spin-ordered phases in the deep Mott insulator regime with anisotropic spin–orbit coupling. We also identify the uniform superfluid, stripe and checkerboard supersolid phases with exotic spin orders in the superfluid regime.


Introduction
The reversible tuning between Mott insulator (MI) and superfluid (SF) phases in ultracold atoms had been realized in the experiment by varying the strength of the periodic lattice potential [1,2]. It is well known that lowenergy properties can be described by the Bose-Hubbard model [3][4][5][6][7][8][9][10][11][12][13][14]. The model provides an ideal platform for the study of MI and SF phases, and the essence of the MI-SF phase transition. In the two-component Bose-Hubbard model, the magnetic phases are predicted in the limit of large repulsive interspecies interactions, such as the incompressible double-checkerboard solid and the supercounterflow [7][8][9]. On the other hand, the possibility of creating artificial gauge fields in the Bose-Einstein condensation (BEC) opens up a new era of the ultracold atom physics [15,16]. Numerous ideas are proposed to generate artificial gauge fields, both Abelian and non-Abelian, with the prediction of many exotic properties in magnetic potentials and in optical lattices as well [17][18][19][20][21][22][23][24][25][26][27][28][29][30].
In this paper, we use a variational order method to explore the phase diagrams of SOC bosons in a square optical lattice. The spin orders of the ground state are non-uniform due to the SOC, so we parameterize each possible order with a proper ansatz function. By comparing the minimized energies of all phases with respect to the variational parameters, we effectively identify the lowest energy states and depict the phase diagram. While the MI-SF phase boundary is computed by the usual Gutzwiller method, the spin-ordered phase structures of the deep MI regime as well as the SF regime are studied by the variational order method. For the isotropic SOC, the spiral (Sp), the vortex crystal (VX) and the skyrmion crystal (SkX) phases are found in the deep MI regime, which is qualitatively similar to the previous results derived by the classical MC simulations, the BDMFT and the spin-wave theory. The Sp phases with different spatial periods are discussed in detail. We further explore the phase diagrams in the MI regime with the anisotropic SOC and in the SF regime. Three new spin-ordered phases, namely, the pseudo-vortex (PVX), the helical (He) and the antiskyrmion (ASkX) crystal phases are foundin the MI regime with the anisotropic SOC. We also identify the uniform superfluid (USF), the stripe supersolid (SSS) and the checkerboard supersolid (CSS) phases with exotic spin orders in the SF regime.
The paper is organized as follows: in section 2 we introduce the Bose-Hubbard model with Rashba-type SOC. In section 3 we compare the MI-SF phase diagram without and with the SOC by using the usual Gutzwiller mean-field theory. The spin-ordered phase diagrams in the deep MI and SF regimes are addressed respectively in section 4.1 and section 4.2 by the variational order method. A brief summary is included in section 5.

Model
The Hamiltonian of the two-species cold bosons in a square optical lattice is written as is the non-Abelian gauge field. The diagonal terms in the matrix denote the spin-conserved tunneling of the bosons whereas the off-diagonal terms denote the Rashbatype SOC which can be generated by a periodic pulsed magnetic field [22,[53][54][55][56]. The two-species bosons with SOC can be rewritten explicitly as  Here å á j i , x (å á j i , y ) represents summation over the neighboring sites in the x(y) direction.
gUare the intra-species and the interspecies interactions, respectively. α and β are the SOC strengths of the x and y directions. In the context, we introduce a parameter l a b = to describe the symmetry of the SOC. The system size is L×L (L = 12) with periodic boundary conditions.

The MI-SF phase transition
In this section, we use the Gutzwiller method to investigate the MI-SF transition with the isotropic SOC (l = 1). The wave function is given by [45]  å   , . The MI-SF phase transition can be determined by using the Gutzwiller method that minimizes the energy functional in equation (4). In figure 1, we display the difference of the phase boundaries without (a = 0 in (a)) and with (a p = 0.2 in (b)) the SOC. The SOC frustrates the kinetic energy and reduces the bandwidth, therefore a larger bare hopping amplitude t is required at larger SOC to reach the SF phase.

Spin-ordered phases
We now investigate the spin-ordered phase diagrams in the deep MI and SF regimes respectively by using thevariational order method. Due to the SOC, the spin-order of the ground state may be non-uniform and hence weparameterize the spin order by an ansatz function for each possible phase. By minimizing the energy functionalwith the variational parameters, we can obtain the lowest energy state as the ground state. In the following, weaddressed the spin-ordered phase diagrams in the deep MI regime in 4.1 and the SF regime in 4.2, respectively. The details of the method are collected in the appendix.   The phase diagrams of the deep MI regime are obtained by minimizing the energy E in equation (7) with respect to all possible macroscopic spin configurations. Due to the competition between the H-type and the DM-type super-exchange couplings, the xy-ferromagnetic (xy-FM), the z-ferromagnetic (z-FM), the zantiferromagnetic (z-AFM), the Sp, the diagonal spiral phase (DSp) (with the spiral vector along the (11) or (1 1) direction), the 2×2 VX and the 3×3 SkX phases are found for the isotropic SOC (l = 1). The results are shown in figure 2(a). In the small α region, the DM-type super-exchange coupling vanishes, the Hamiltonian reduces to the standard XXZ-Heisenberg model, where the H-type super-exchange coupling of the x−y inplane component is larger than the z-component for < g 1, and vice verse. In the limit a p  2, the case is identical to the small α. The H-type super-exchange along the x and the y directions are of different signs which lead to the H-type super-exchange in spin-space has different sign for the x and y components and form the VX phase. The DM-type super-exchange coupling grows and the coplanar state becomes unstable, giving way to the SkX phases as α reduces. The Sp phase results from the combination of the H-type super-exchange and the nonzero DM-type super-exchange in the intermediate values of SOC. Though the phase boundaries are not match well, the phase diagram is qualitatively similar to the previous result derived by the classical MC simulations, as the figure 2(a) in [25]. The various Sp phases with different spatial periods of n lattice sites (   n 3 12) (called the Sp-n phase) are explicitly displayed in figure 2(b). We note that the spatial periods decrease with the increase of the SOC strength. Figure 3 shows the ground state phase diagrams for the anisotropic SOC (l ¹ 1). In contrast to the isotropic SOC, the 3×3 SkX and the Sp phases are replaced by the 3×3 ASkX and the 2×2 PVX phases for l = 0.5. Here, the ASkX phase denotes that the spin at the center points to negative z direction while the spins away from the center tumble outward the positive z direction. The PVX phase denotes a non-coplanar state with q ¹  at the even number lattice sites pose a spiral pattern for l = 5. Due to the symmetry of DM-type super-exchange and the coplanar feature of the system are broken, the 3×3 ASkX, the 2×2 PVX and the He phases are found in the anisotropic SOC system.

SF regime
In the SF regime, the wave functions of the two-component bosons are characterized in terms of the four parameters (r f q j , , , where r j and f j are the density and phase at site j with normalization r å = 1 j j . Substituting equation (8) into equation (2) gives rise to the energy functional The ground state phase diagrams of the SF regime are obtained by minimizing the E in equation (9) with respect to various configurations of the parameters (r f q j , , , j j j j ). For the isotropic SOC, four types of phases are found in figure 4. These phases are the USF phase with xy-FM order (USF-xy-FM), the USF phase with Sp order with spatial period of 3 lattice sites (USF-Sp-3), the SSS phase with Sp order with spatial period of 12 lattice sites (SSS-Sp-12), and the USF phase with 2×2 spin VX order (USF-2 × 2-VX). Figure 5 displays the spin-ordered phase diagrams for the anisotropic SOC system. Except the phases revealed in figure (4), we observe the SSS phases with 2×2 spin VX order (SSS-2 × 2-VX), the xy-FM order (SSS-xy-FM), the He order with spatial period of 12 lattice sites (SSS- , and the USF phase with 3×3 SkX order (USF-3 × 3-SkX). In figure 5(c), a CSS phase of He order with spatial period of 12 lattice sites (CSS-  appears. In certain regions, the USF, the SSS and the CSS phases with exotic spin orders are preferred since they have the lowest energies. (d) is the enlarged part of the red dashed region of (c). The blue fonts indicate the new found spin-ordered phases, i.e., the PVX, the He and the ASkX phases.

Summary
We have used a variational order method to study the ground state phase diagrams in the deep MI and the SF regimes for the SOC bosons in the optical lattice. In the deep MI regime, the Sp, the VX and the SkX phases are found for the isotropic SOC system. Three new spin-ordered phases, the PVX, the He and the ASkX phases appeared for the anisotropic SOC system. We also identify the USF, the SSS and the CSS phases with exotic spinorder in the SF regime. Our method is effective to deal with such systems and can provide clues to search the various magnetic phases in the ultracold atom experiment. The method also can be extended to other optical lattices and to higher dimensional spaces.

Appendix. Variational method
In the deep MI regime, the parameters (q j , j j ) can be parameterized by the a distribution function. For example, in the Sp-n phase we suppose that     The energy reaches the minimum value as the variational parameters a = p 2 and b = = p n , 3 4 , which is the 3×3 SkX phase. All other phases in the deep MI regime are obtained by these processes. The phase diagrams in the SF regime are obtained by similar processes.