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arXiv:1205.3116 · Physical Review A

Magnetic phases of bosons with synthetic spin-orbit coupling in optical lattices

Zi Cai, Xiang-Fa Zhou, Congjun Wu

Published Jun 20, 2012• 98 citations
arXivHTMLPDF

Abstract

We investigate magnetic properties in the superfluid and Mott-insulating states of two-component bosons with spin-orbit (SO) coupling in two-dimensional square optical lattices. The spin-independent hopping integral $t$ and SO coupled one $\ensuremath{\lambda}$ are fitted from band-structure calculations in the continuum, which exhibit oscillations with increasing SO coupling strength. The magnetic superexchange model is derived in the Mott-insulating state with one particle per site, characterized by the Dzyaloshinsky-Moriya interaction. In the limit of $|\ensuremath{\lambda}|\ensuremath{\ll}|t|$, we find a spin spiral Mott state whose pitch value is the same as that in the incommensurate superfluid state, while in the opposite limit $|t|\ensuremath{\ll}|\ensuremath{\lambda}|$, the ground state exhibits a $2\ifmmode\times\else\texttimes\fi{}2$ in-plane spin pattern.

DOI
10.1103/physreva.85.061605

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We investigate magnetic properties in the superfluid and Mott-insulating states of two-component bosons with spin-orbit (SO) coupling in two-dimensional square optical lattices. The spin-independent hopping integral ttt and SO coupled one \ensuremathλ\ensuremath{\lambda}\ensuremathλ are fitted from band-structure calculations in the continuum, which exhibit oscillations with increasing SO coupling strength. The magnetic superexchange model is derived in the Mott-insulating state with one particle per site, characterized by the Dzyaloshinsky-Moriya interaction. In the limit of ∣\ensuremathλ∣\ensuremath≪∣t∣|\ensuremath{\lambda}|\ensuremath{\ll}|t|∣\ensuremathλ∣\ensuremath≪∣t∣, we find a spin spiral Mott state whose pitch value is the same as that in the incommensurate superfluid state, while in the opposite limit ∣t∣\ensuremath≪∣\ensuremathλ∣|t|\ensuremath{\ll}|\ensuremath{\lambda}|∣t∣\ensuremath≪∣\ensuremathλ∣, the ground state exhibits a 2\ifmmode×\else\texttimes\fi22\ifmmode\times\else\texttimes\fi{}22\ifmmode×\else\texttimes\fi2 in-plane spin pattern.
We investigate the ground-state properties of Bose-Bose mixtures with Rashba-type spin-orbit (SO) coupling in a square lattice. The system displays rich physics from the deep Mott insulator (MI) all the way to the superfluid (SF) regime. In the deep MI regime, exotic spin-ordered phases arise due to the effective Dzyaloshinskii-Moriya type of superexchange interactions. By employing the nonperturbative bosonic dynamical mean-field theory (BDMFT), we numerically study and establish the stability of these magnetic phases against increasing hopping amplitude. We show that as hopping is increased across the MI to SF transition, exotic superfluid phases with magnetic textures emerge. In particular, we identify an exotic spin-spiral magnetic texture with spatial period 3 in the superfluid close to the MI-SF transition.
We study ultracold bosonic atoms with synthetic three-dimensional spin-orbit (SO) coupling in a cubic optical lattice. In the superfluidity phase, the lowest energy band exhibits one, two, or four pairs of degenerate single-particle ground states depending on the SO-coupling strengths, which can give rise to condensate states with spin stripes for weak atomic interactions. In the deep Mott-insulator regime, the effective spin Hamiltonian of the system combines three-dimensional Heisenberg exchange interactions, anisotropy interactions, and Dzyaloshinskii-Moriya interactions. Based on Monte Carlo simulations, we numerically demonstrate that the resulting Hamiltonian with an additional Zeeman field has a rich phase diagram with spiral, stripe, vortex crystal, and especially Skyrmion crystal spin textures in each xyxyxy-plane layer. The obtained Skyrmion crystals can be tunable with square and hexagonal symmetries in a columnar manner along the zzz axis, and moreover are stable against the interlayer spin-spin interactions in a large parameter region.
We study the Mott phase of three-component bosons, with one particle per site, in an optical lattice by mapping it onto an SU(3) spin model. In the simplest case of full SU(3) symmetry, one obtains a ferromagnetic Heisenberg model. Introducing an SU(3) analog of spin-orbit coupling, additional spin-spin interactions are generated. We first consider the scenario of spin-dependent hopping phases, leading to Dzyaloshinskii-Moriya-type interactions. They result in the formation of spiral spin textures, which in one dimension can be understood by a local unitary transformation. Applying classical Monte Carlo simulations, we extend our study to two-dimensional systems, and systems with ``true'' spin-orbit coupling, involving spin-changing hoppings.
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 study the ground state properties of spin-1 bosons in a two-dimensional optical lattice, by applying a variational Monte Carlo method to the S=1 Bose-Hubbard model on a square lattice at unit filling. A doublon-holon binding factor introduced in the trial state provides a noticeable improvement in the variational energy over the conventional Gutzwiller wave function and allows us to deal effectively with the inter-site correlations of particle densities and spins. We systematically show how spin-dependent interactions modify the superfluid-Mott insulator transitions in the S=1 Bose-Hubbard model due to the interplay between the density and spin fluctuations of bosons. Furthermore, regarding the magnetic phases in the Mott region, the calculated spin structure factor elucidates the emergence of nematic and ferromagnetic spin orders for antiferromagnetic (U2>0U_2>0U2​>0) and ferromagnetic (U2<0U_2<0U2​<0) couplings, respectively.
We study, using quantum Monte Carlo (QMC) simulations, the ground state properties of spin-1 bosons trapped in a square optical lattice. The phase diagram is characterized by the mobility of the particles (Mott insulating or superfluid phase) and by their magnetic properties. For ferromagnetic on-site interactions, the whole phase diagram is ferromagnetic and the Mott insulators-superfluid phase transitions are second order. For antiferromagnetic on-site interactions, spin nematic order is found in the odd Mott lobes and in the superfluid phase. Furthermore, the superfluid-insulator phase transition is first or second order depending on whether the density in the Mott is even or odd. Inside the even Mott lobes we observe a singlet-to-nematic transition for certain values of the interactions. This transition appears to be first order.