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.
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.
We consider a square optical lattice in two dimensions and study the effects of both the strength and symmetry of spin-orbit coupling and Zeeman field on the ground-state, i.e., Mott-insulator (MI) and superfluid (SF), phases and phase diagram, i.e., MI-SF phase-transition boundary, of the two-component Bose-Hubbard model. In particular, based on a variational Gutzwiller ansatz, our numerical calculations show that the spin-orbit-coupled SF phase is a nonuniform (twisted) one, with its phase (but not the magnitude) of the order parameter modulating from site to site. Fully analytical insights into the numerical results are also given.
Motivated by recent experimental advances, we study two-component spin-orbit coupled ultracold bosonic atoms in two dimensions on a square optical lattice. Using a Bose-Hubbard model with spin-conserving and non-spin-conserving nearest neighbour hoppings and spin-dependent on-site density-density interaction, our goal is to characterize phase separation and spin structure in the weakly interacting and deep Mott regimes. \nFor the weakly coupled regime, we decouple interactions through a real space uniform density mean field theory. At zero temperature, this gives an analytic condition for the phase separation transition driven by inter- relative to intracomponent interaction. Solving the self-consistent equations at finite temperature reveals entropic remixing in the phase separated regime and a more surprising entropy driven phase separation in the mixed regime. This is a consequence of complex interplay between interaction and spin-orbit coupling, and can be explained through the effect of component imbalance on the effective single-particle dispersion relation. We also provide an alternate explanation based on thermal occupation of eigenstates with a characteristic imbalance.\nIn the strongly interacting Mott regime, we derive an effective spin Hamiltonian describing the magnetic phases of the Mott insulator. The competition between anisotropic {Heisenberg} and Dzyaloshinskii-Moriya interactions gives rise to various ferromagnetic, antiferromagnetic, spiral, stripe, vortex, and skyrmion phases. On basis of classical Monte-Carlo simulations in the literature, we reconstruct the phase diagram with a classical variational approach, while magnon excitation spectra and quantum fluctuations are calculated with Holstein-Primakoff transformation and subsequent spin wave expansion. The analysis shows that states with ferromagnetic or antiferromagnetic ordering of boson species are protected against thermal fluctuations by a gap, and well described by classical states. States with equal superposition of boson species at each lattice site are subject to relatively large quantum fluctuations, which may cause breakdown of the states within their classical parameter space regions. The dispersion relations are gapless and linear around the minima to lowest order in the spin wave expansion.
Mott insulator to superfluid phase transitions and unconventional superfluid phases in spin-orbit coupled Bose-Einstein condensate in one dimensional, square and hexagonal optical lattices are investigated using decoupling approximation and variational Gutzwiller wave function. We considered a system with two components. In the first part, we used decoupling approximation along side perturbation to chart the phase diagrams of the system in the absence of spin-orbit coupling. Our results show that the occupation number of the species, interspecies interaction and the ratio of the hopping matrix of the species dictate the critical point and the separation of the insulating phases. Applying decoupling approximation to spin-orbit coupled Bose-Einstein condensate in one dimensional optical we find that spin-orbit coupling reduces the phase boundary of Mott insulator-superfluid transition and also modifies the critical point in a way that the critical point reduces as the coupling strength increases. In the second part, we applied the variational Gutzwiller wave function to spin-orbit coupled Bose-Einstein condensate in square and hexagonal lattice. Our results show that the geometry of the optical lattice plays a role in the transition. We find that the mean-field Mott insulator-superfluid phase transition boundary of hexagonal lattice is smaller than that of square lattice for the same system. Finally, we showed that the Gutzwiller variational approach gives us access to the twisted superfluid phase realized in the system. The nonuniformity of the superfluid phase is different for the two lattices.
We study cold atoms in an optical lattice with synthetic spin-orbit coupling in the Mott-insulator regime. We calculate the parameters of the corresponding tight-binding model using Peierls substitution and "localized Wannier states method" and derive the low-energy spin Hamiltonian for bosons and fermions. The spin Hamiltonian is a combination of Heisenberg model, quantum compass model and Dzyaloshinskii-Moriya interactions and it has a rich classical phase diagram with collinear, spiral and vortex phases. We discuss the state of the art of experiments to realize and detect magnetic orderings in strongly correlated 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, 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, 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, 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.
We study the physics of interacting spin-1 bosons in an optical lattice using a variational Gutzwiller technique. We compute the mean-field ground state wave function and discuss the evolution of the condensate, spin, nematic, and singlet order parameters across the superfluid-Mott transition. We then extend the Gutzwiller method to derive the equations governing the dynamics of low energy excitations in the lattice. Linearizing these equations, we compute the excitation spectra in the superfluid and Mott phases for both ferromagnetic and antiferromagnetic spin-spin interactions. In the superfluid phase, we recover the known excitation spectrum obtained from Bogoliubov theory. In the nematic Mott phase, we obtain gapped, quadratically dispersing particle and hole-like collective modes, whereas in the singlet Mott phase, we obtain a nondispersive gapped mode, corresponding to the breaking of a singlet pair. For the ferromagnetic Mott insulator, the Gutzwiller mean-field theory only yields particle-hole-like modes but no Goldstone mode associated with long-range spin order. To overcome this limitation, we supplement the Gutzwiller theory with a Schwinger boson mean-field theory which captures superexchange-driven fluctuations. In addition to the gapped particle-hole-like modes, we obtain a gapless quadratically dispersing ferromagnetic spin-wave Goldstone mode. We discuss the evolution of the singlet gap, particle-hole gap, and the effective mass of the ferromagnetic Goldstone mode as the superfluid-Mott phase boundary is approached from the insulating side. We discuss the relevance and validity of Gutzwiller mean-field theories to spinful systems, and potential extensions of this framework to include more exotic physics which appears in the presence of spin-orbit coupling or artificial gauge fields.