Finite-momentum superfluid and supersolid phases in a spin-orbit-coupled spin-1 Bose-Hubbard model with nearest-neighbor interaction

Abstract

The interplay between spin-orbit coupling (SOC) and nearest-neighbor (NN) repulsion in optical lattices provides a fertile ground for realizing exotic quantum phases that simultaneously host finite-momentum condensation and spontaneous density modulation. We address this interplay for spin-1 bosons in a two-dimensional square lattice with Rashba SOC and NN interactions, employing the single-site Gutzwiller mean-field (SGMF) method as our primary tool and the $2\times2$ cluster Gutzwiller mean-field (CGMF) approach as a systematic check of short-range correlation effects. Our central finding is that SOC fundamentally reconstructs the single-particle band structure, shifting the lowest-band minimum from zero to finite momenta with increasing SOC strength. This reconstruction underpins a cascade of modulated superfluid (SF) phases-stripe, square, twist, and checkerboard-whose condensate textures are dictated predominantly by SOC. The addition of NN repulsion naturally favors real-space density ordering; the competition and cooperation between this density order and SOC-generated phase textures then stabilize a family of supersolid (SS) phases that mirror the same four modulated patterns, in which finite-momentum condensate and density-wave orders coexist. To complement the numerical phase diagrams, we derive semi-analytical phase boundaries for both the Mott-insulator (MI)-SF and density-wave (DW)-SS transitions using second-order perturbation theory. These expressions not only reproduce the overall trends of the SGMF results but also provide a transparent physical picture of the underlying instability mechanisms. Importantly, the representative CGMF calculations retain the same DW lobes and intervening SS regions, albeit with quantitative shifts, while systematic finite-size analyses reveal only weak dependence on lattice size, with no evidence of phase separation, collectively attesting to the robustness of the predicted phase structure. Together, our results establish a comprehensive mean-field framework that unifies the SOC-driven finite-momentum condensation and the NN-interaction-induced density ordering within a single picture, offering a valuable benchmark for future experimental explorations of spin-orbit-coupled spinor gases in optical lattices.