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arXiv:1509.00005 · Physical review. B./Physical review. B

Interaction-driven exotic quantum phases in spin-orbit-coupled spin-1 bosons

J. H. Pixley, Stefan S. Natu, Ian B. Spielman, S. Das Sarma

Published Feb 1, 2016• 14 citations
arXivHTMLPDF

Abstract

We study the interplay between large-spin, spin-orbit coupling, and superfluidity for bosons in a two-dimensional optical lattice, focusing on the spin-1 spin-orbit-coupled system recently realized at the Joint Quantum Institute [Campbell et al., arXiv:1501.05984]. We find a rich quantum phase diagram where, in addition to the conventional phases---superfluid and insulator---contained in the spin-1 Bose-Hubbard model, there are new lattice symmetry breaking phases. For weak interactions, the interplay between two length scales, the lattice momentum and the spin-orbit wave vector, induce a phase transition from a uniform superfluid to a phase where bosons simultaneously condense at the center and edge of the Brillouin zone at a nonzero spin-orbit strength. This state is characterized by spin-density-wave order, which arises from the spin-1 nature of the system. Interactions suppress spin-density-wave order, and favor a superfluid only at the Brillouin zone edge. This state has spatially oscillating mean-field order parameters, but a homogeneous density. We show that the spin-density-wave superfluid phase survives in a two-dimensional harmonic trap, and thus establish that our results are directly applicable to experiments on $^{87}\mathrm{Rb},\phantom{\rule{0.16em}{0ex}}^{7}\mathrm{Li}$, and $^{41}\mathrm{K}$.

DOI
10.1103/physrevb.93.081101

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We study the interplay between large-spin, spin-orbit coupling, and superfluidity for bosons in a two-dimensional optical lattice, focusing on the spin-1 spin-orbit-coupled system recently realized at the Joint Quantum Institute [Campbell et al., arXiv:1501.05984]. We find a rich quantum phase diagram where, in addition to the conventional phases---superfluid and insulator---contained in the spin-1 Bose-Hubbard model, there are new lattice symmetry breaking phases. For weak interactions, the interplay between two length scales, the lattice momentum and the spin-orbit wave vector, induce a phase transition from a uniform superfluid to a phase where bosons simultaneously condense at the center and edge of the Brillouin zone at a nonzero spin-orbit strength. This state is characterized by spin-density-wave order, which arises from the spin-1 nature of the system. Interactions suppress spin-density-wave order, and favor a superfluid only at the Brillouin zone edge. This state has spatially oscillating mean-field order parameters, but a homogeneous density. We show that the spin-density-wave superfluid phase survives in a two-dimensional harmonic trap, and thus establish that our results are directly applicable to experiments on 87Rb,7Li^{87}\mathrm{Rb},\phantom{\rule{0.16em}{0ex}}^{7}\mathrm{Li}87Rb,7Li, and 41K^{41}\mathrm{K}41K.
Ultracold bosons in optical lattices are one of the few systems where bosonic matter is known to exhibit strong correlations. Here we push the frontier of our understanding of interacting bosons in optical lattices by adding synthetic spin-orbit coupling, and show that new kinds of density- and chiral-orders develop. The competition between the optical lattice period and the spin-orbit coupling length - which can be made comparable in experiments - along with the spin hybridization induced by a transverse field (i.e., Rabi coupling) and interparticle interactions create a rich variety of quantum phases including uniform, non-uniform and phase-separated superfluids, as well as Mott insulators. The spontaneous symmetry breaking phenomena at the transitions between them are explained by a two-order-parameter Ginzburg-Landau model with multiparticle umklapp processes. Finally, in order to characterize each phase, we calculated their experimentally measurable crystal momentum distributions. PACS numbers: 67.85.-d,67.85.Hj,67.85.Fg.
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 2×22\times22×2 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.
In recent experiments, spin-orbit-coupled (SOC) bosonic gases in an optical lattice have been successfully prepared into any Bloch band [Hamner et al., Phys. Rev. Lett. 114, 070401 (2015)], which promises a viable contender in the competitive field of simulating gauge-related phenomena. However, the ground-state phase diagram of such systems in the superfluid regime is still lacking. Here we present a detailed study of the phase diagram in an optically trapped Bose gas with equal-weight Rashba and Dresselhaus SO coupling. We identify four different quantum phases, which include three normal phases and a mixed phase, by considering the wave vector k1{k}_{1}k1​, the longitudinal $\ensuremath{\langle}{\ensuremath{\sigma}}_{z}\ensuremath{\rangle}$, and the transverse $\ensuremath{\langle}{\ensuremath{\sigma}}_{x}\ensuremath{\rangle}$ spin polarizations as three order parameters. The ground state of normal phases is a Bloch wave with a single wave vector k1{k}_{1}k1​, which can position in arbitrary regions in the Brillouin zone. By contrast, the ground state of the mixed phase is a superposition of two Bloch waves with opposite k1{k}_{1}k1​, which, remarkably, may lack periodicity even though the system's Hamiltonian is periodic. This mixed phase in the lattice setting can be seen as the counterpart of the stripe phase associated with the uniform SOC gas. Furthermore, due to the lattice-renormalized SOC, the phase diagram of the model system becomes significantly different from the uniform case when the lattice strength grows. Finally, a scheme for experimentally probing the mixed phase using Bragg spectroscopy is proposed.
Motivated by recent experiments [Y. J. Lin et al., Nature (London) 471, 83 (2011)], we study Mott phases and superfluid-insulator (SI) transitions of two-species ultracold bosonic atoms in a two-dimensional square optical lattice with nearest-neighbor hopping amplitude ttt and in the presence of a spin-orbit coupling characterized by a tunable strength $\ensuremath{\gamma}$. Using both strong-coupling expansion and Gutzwiller mean-field theory, we chart out the phase diagrams of the bosons in the presence of such spin-orbit interaction. We compute the momentum distribution of the bosons in the Mott phase near the SI transition point and show that it displays precursor peaks whose position in the Brillouin zone can be varied by tuning $\ensuremath{\gamma}$. Our analysis of the critical theory of the transition unravels the presence of unconventional quantum critical points at $t/\ensuremath{\gamma}=0$, which are accompanied by emergence of an additional gapless mode in the critical region. We also study the superfluid phases of the bosons near the SI transition using a Gutzwiller mean-field theory that reveals the existence of a twisted superfluid phase with an anisotropic twist angle which depends on $\ensuremath{\gamma}$. Finally, we compute the collective modes of the bosons and point out the presence of reentrant SI transitions as a function of $\ensuremath{\gamma}$ for nonzero ttt. We propose experiments to test our theory.
We investigate the quantum phase transitions of a two-dimensional Bose-Hubbard model in the presence of a Rashba spin-orbit coupling with and without thermal fluctuations. The interplay of single-particle hopping, strength of spin-orbit coupling, and interspin interaction leads to superfluid phases with distinct properties. With interspin interactions weaker than intraspin interactions, the spin-orbit coupling induces two finite-momentum superfluid phases. One of them is a phase-twisted superfluid that exists at low hopping strengths and reduces the domain of insulating phases. At comparatively higher hopping strengths, there is a transition from the phase-twisted to a finite-momentum stripe superfluid. With interspin interactions stronger than the intraspin interactions, the system exhibits a phase-twisted to ferromagnetic phase transition. At finite temperatures, the thermal fluctuations destroy the phase-twisted superfluidity and lead to a wide region of normal-fluid states. These findings can be observed in recent quantum gas experiments with spin-orbit coupling in optical lattices.
We investigate the thermal physics of a Bose-Hubbard model with Rashba spin-orbit coupling starting from a strong coupling mean-field ground state. The essential role of the spin-orbit coupling $\left(\ensuremath{\gamma}\right)$ is to promote condensation of the bosons at a finite wave vector ${\mathbit{k}}_{0}$. We find that the bosons display either homogeneous or phase-twisted or orbital ordered superfluid phases, depending on $\ensuremath{\gamma}$ and the interspecies interaction strength $\left(\ensuremath{\lambda}\right)$. We show that an increase of $\ensuremath{\gamma}$ leads to suppression of the critical interaction Uc{U}_{c}Uc​ for the superfluid to Mott insulator transition in the ground state and a reduction of the Tc{T}_{c}Tc​ for superfluid to Bose-liquid transition at a fixed interaction strength. We capture the thermal broadening in the momentum distribution function, and the real space profiles of the thermally disordered magnetic textures, including their homogenization for $T\ensuremath{\gtrsim}{T}_{c}$. We provide a Landau theory based description of the ground state phase boundaries and thermal transition scales and discuss experiments which can test our theory.