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arXiv:1409.1216 · Physical Review Letters

Thermal Phase Transitions of Strongly Correlated Bosons with Spin-Orbit Coupling

Ciarán Hickey, Arun Paramekanti

Published Dec 24, 2014• 30 citations
arXivHTMLPDF

Abstract

Experiments on ultracold atoms have started to explore lattice effects and thermal fluctuations for two-component bosons with spin-orbit coupling (SOC). Motivated by this, we derive and study a tJ model for lattice bosons with equal Rashba-Dresselhaus SOC and strong Hubbard repulsion in a uniform Zeeman magnetic field. Using the Gutzwiller ansatz, we find strongly correlated ground states with stripe superfluid (SF) order. We formulate a finite temperature generalization of the Gutzwiller method, and show that thermal fluctuations in the doped Mott insulator drive a two-step melting of the stripe SF, revealing a wide regime of a stripe normal fluid.

DOI
10.1103/physrevlett.113.265302

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Experiments on ultracold atoms have started to explore lattice effects and thermal fluctuations for two-component bosons with spin-orbit coupling (SOC). Motivated by this, we derive and study a tJ model for lattice bosons with equal Rashba-Dresselhaus SOC and strong Hubbard repulsion in a uniform Zeeman magnetic field. Using the Gutzwiller ansatz, we find strongly correlated ground states with stripe superfluid (SF) order. We formulate a finite temperature generalization of the Gutzwiller method, and show that thermal fluctuations in the doped Mott insulator drive a two-step melting of the stripe SF, revealing a wide regime of a stripe normal fluid.
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.
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.
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.
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.
In this Rapid Communication we study finite-temperature properties of spin-1/21/21/2 interacting bosons with spin-orbit coupling in two dimensions. When the ground state has stripe order, we show that thermal fluctuations will first melt the stripe order and lead to a superfluid of boson pairs if the spin-orbit coupling is isotropic or nearly isotropic. Such a phase supports fractionalized quantum vortices. The Kosterlitz-Thouless transition from superfluid to normal state is driven by proliferation of half vortices. When the ground state is a plane-wave state, the transition to normal state is driven by conventional Kosterlitz-Thouless transition. However, the critical temperature will drop to zero for isotropic spin-orbit coupling.
Motivated by the experimental realization of synthetic spin-orbit coupling for ultracold atoms, we investigate the phase diagram of the Bose-Hubbard model in a non-Abelian gauge field in two dimensions. Using a strong coupling expansion in the combined presence of spin-orbit coupling and tunable interactions, we find a variety of interesting magnetic Hamiltonians in the Mott insulator (MI), which support magnetic textures such as spin spirals and vortex and Skyrmion crystals. An inhomogeneous mean-field treatment shows that the superfluid (SF) phases inherit these exotic magnetic orders from the MI and display, in addition, unusual modulated current patterns. We present a slave-boson theory which gives insight into such intertwined spin-charge orders in the SF, and discuss signatures of these orders in Bragg scattering, in situ microscopy, and dynamic quench experiments.