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arXiv:1601.01642 · New Journal of Physics

Fermionic Hubbard model with Rashba or Dresselhaus spin–orbit coupling

Fadi Sun, Jinwu Ye, Wu-Ming Liu

Published Apr 20, 2017• 14 citations
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Abstract

In this work, we investigate the possible dramatic effects of Rashba or Dresselhaus spin–orbit coupling (SOC) on the fermionic Hubbard model in a two-dimensional square lattice. In the strong coupling limit, it leads to the rotated antiferromagnetic Heisenberg model which is a new class of quantum spin model. For a special equivalent class, we identify a new spin–orbital entangled commensurate ground (Y-y) state subject to strong quantum fluctuations at T = 0. We evaluate the quantum fluctuations by the spin wave expansion up to order . In some SOC parameter regimes, the Y-y state supports a massive relativistic incommensurate magnon (C-IC) with its two gap minima positions continuously tuned by the SOC parameters. The C-IC magnons dominate all the low temperature thermodynamic quantities and also lead to the separation of the peak positions between the longitudinal and the transverse spin structure factors. In the weak coupling limit, any weak repulsive interaction also leads to a weak Y-y state. There is only a crossover from the weak to the strong coupling. High temperature expansions of the specific heats in both weak and strong coupling are presented. The dramatic roles to be played by these C-IC magnons at generic SOC parameters or under various external probes are hinted at. Experimental applications to both layered noncentrosymmetric materials and cold atoms are discussed.

DOI
10.1088/1367-2630/aa6e31

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In this work, we investigate the possible dramatic effects of Rashba or Dresselhaus spin–orbit coupling (SOC) on the fermionic Hubbard model in a two-dimensional square lattice. In the strong coupling limit, it leads to the rotated antiferromagnetic Heisenberg model which is a new class of quantum spin model. For a special equivalent class, we identify a new spin–orbital entangled commensurate ground (Y-y) state subject to strong quantum fluctuations at T = 0. We evaluate the quantum fluctuations by the spin wave expansion up to order . In some SOC parameter regimes, the Y-y state supports a massive relativistic incommensurate magnon (C-IC) with its two gap minima positions continuously tuned by the SOC parameters. The C-IC magnons dominate all the low temperature thermodynamic quantities and also lead to the separation of the peak positions between the longitudinal and the transverse spin structure factors. In the weak coupling limit, any weak repulsive interaction also leads to a weak Y-y state. There is only a crossover from the weak to the strong coupling. High temperature expansions of the specific heats in both weak and strong coupling are presented. The dramatic roles to be played by these C-IC magnons at generic SOC parameters or under various external probes are hinted at. Experimental applications to both layered noncentrosymmetric materials and cold atoms are discussed.
In this work, we study strongly interacting spinor atoms in a lattice subject to a two dimensional (2d) anisotropic Rashba type of spin orbital coupling (SOC) and an Zeeman field. We find the interplay between the Zeeman field and the SOC provides a new platform to host rich and novel classes of quantum commensurate and in-commensurate phases, excitations and phase transitions. These commensurate phases include two collinear states at low and high Zeeman field, two co-planar canted states at mirror reflected SOC parameters respectively. Most importantly, there are non-coplanar incommensurate Skyrmion (IC-SkX) crystal phases surrounded by the four commensurate phases. New excitation spectra above all the five phases, especially on the IC-SKX phase are computed. Three different classes of quantum commensurate to in-commensurate transitions from the IC-SKX to its four neighboring commensurate phases are identified. Finite temperature behaviors and transitions are discussed. The critical temperatures of all the phases can be raised above that reachable by current cold atom cooling techniques simply by tuning the number of atoms N per site. In view of recent impressive experimental advances in generating 2d SOC for cold atoms in optical lattices, these new many-body phenomena can be explored in the current and near future cold atom experiments. Applications to various materials such as MnSi, Fe 0.5 Co 0.5 Si, especially the complex incommensurate magnetic ordering in Li 2 IrO 3 are given.
We study quantum magnetism of interacting spinor bosons at integer fillings hopping in a square lattice in the presence of non-Abelian gauge fields. In the strong-coupling limit, this leads to the rotated ferromagnetic Heisenberg model, which is a new class of quantum spin model. We introduce Wilson loops to characterize frustrations and gauge equivalent classes. For a special equivalent class, we identify a spin-orbital entangled commensurate ground state. It supports not only commensurate magnons, but also a gapped elementary excitation: incommensurate magnons with two gap minima continuously tuned by the spin-orbit coupling (SOC) strength. At low temperatures, these magnons lead to dramatic effects in many physical quantities such as density of states, specific heat, magnetization, uniform susceptibility, staggered susceptibility, and various spin-correlation functions. The commensurate magnons lead to a pinned central peak in the angle-resolved light or atom Bragg spectroscopy. However, the incommensurate magnons split it into two located at their two gap minima. At high temperatures, the transverse spin-structure factors depend on the SOC strength explicitly. The whole set of Wilson loops can be mapped out by measuring the specific heat at the corresponding orders in the high-temperature expansion. We argue that one gauge may be realized in current experiments and other gauges may also be realized in future experiments. The results achieved along the exact solvable line sets up the stage to investigate dramatic effects when tuning away from it by various means. We sketch the crucial roles to be played by these magnons at other equivalent classes, with spin anisotropic interactions and in the presence of finite magnetic fields. Various experimental detections of these phenomena are discussed.
In a previous work, we derived a Rotated Ferromagnetic Heisenberg model (RFHM) to describe quantum magnetic phenomena of spinor bosons loaded in optical lattices in the presence of spin-orbit coupling (SOC). In this paper, we study the RFHM at a special equivalent class in a Zeeman field. We find that the interplay of the Zeeman field and the SOC on a lattice leads to rich and new quantum phases through new universality class of quantum phase transitions. These quantum phases include two collinear states at low and high Zeeman field, two co-planar canted states at two dual related SOC strengths respectively, also a non-coplanar incommensurate Skyrmion (IC-SkX) crystal phases surrounded by the 4 phases. The IC-SkX state reduces to a commensurate SkX state with a 2X4 spin-orbital structure at a special commensurate value of the SOC. We determine the orbital and spin structures and the novel excitation spectra of all the quantum phases. We also explore new universality classes of quantum phase transitions between (among) these phases with the dynamic exponents z=1, z=2 and the anisotropic one (zx=1z_x = 1zx​=1, zy=3z_y = 3zy​=3) and finite temperature transitions above all these phases. Experimental implementations and various experimental detections of these new phenomena are also discussed. Possible implications to condensed matter systems with or without SOC are given.
In this advance, we study the system of strongly interacting spinor bosons in a square lattice subject to any of the linear combinations of the Rashba and Dresselhaus spin-orbit coupling (SOC). The SOC leads to novel and rich quantum and topological phenomena in a lattice system. They include masses generated from order from quantum disorder mechanism, collinear spin-bond correlated magnetic phase, quantum commensurate ( C ) and In-commensurate ( IC ) non-coplanar Skyrmion crystal phases, stripe co-planar ( spiral ) C and IC phases, quantum Lifshitz C-IC transitions, metastable states, hysteresis, topological rational and irrational winding numbers, incomplete and complete devil staircases, Cantor sets, multi-fractals, etc. Various perspectives and possible intimate connections with topological spin liquids due to geometric frustrations or quantum spin glass due to quenched disorders are discussed. Implications on current or near future cold atom systems and on 4d or 5d strongly correlated materials with SOC are discussed.
We clarify the ground-state phase diagram of the half-filled square-lattice Hubbard model with Rashba spin-orbit coupling (SOC) characterized by the spin-split energy bands due to broken inversion symmetry. Although the Rashba metals and insulating magnets have been studied well, the intermediate interaction strength of the system remained elusive due to the lack of appropriate theoretical tools to unbiasedly describe the large-scale magnetic structures. We complementarily apply four different methods (sine-square deformed mean-field theory, random phase approximation, Luttinger-Tisza method, and density matrix embedding theory) and succeed in capturing the incommensurate spin-density-wave (SDW) phases with very long spatial periods which were previously overlooked. The transition to the SDW phases from the metallic phase is driven by an unprecedented instability that nests the two parts of the Fermi surface carrying opposite spins. For large SOC, the spiral, stipe, and vortex phases are obtained, when the four Dirac points exist near the Fermi level and their whole linear dispersions nest by a wavelength $\ensuremath{\pi}$, opening a band gap. These two types of transition provide Fermiology that distinguishes the antisymmetric SOC systems, generating a variety of magnetic phases that start from the relatively weak correlation regime and continue to the strongly interacting limit.
In this paper, we study the rotated ferromagnetic Heisenberg model (RFHM) in two different transverse fields, hx{h}_{x}hx​ and hz{h}_{z}hz​, which can be intuitively visualized as studying spin-orbit coupling (SOC) effects in two-dimensional (2D) Ising or anisotropic XYXYXY model in a transverse field. At a special SOC class, it was found in our previous work [Phys. Rev. A 92, 043609 (2015)] that the RFHM at a zero field owns an exact spin-orbit coupled ground state called the $Y\text{\ensuremath{-}}x$ state. It supports not only the commensurate magnons (called $\mathrm{C}\text{\ensuremath{-}}{\mathrm{C}}_{0}$ and $\mathrm{C}\text{\ensuremath{-}}{\mathrm{C}}_{\ensuremath{\pi}})$, but also the incommensurate magnons (called C-IC). These magnons are nonrelativistic, not embedded in the exact ground state, so need to be thermally excited or generated by various external probes. Their dramatic response under a longitudinal hy{h}_{y}hy​ field was recently worked out by Sun et al. [arXiv:1502.05338]. Here we find they respond very differently under the two transverse fields. Any hx{h}_{x}hx​ (hz)({h}_{z})(hz​) introduces quantum fluctuations to the ground state and changes the collinear $Y\text{\ensuremath{-}}x$ state to a canted coplanar $YX\text{\ensuremath{-}}x$ $(YZ\text{\ensuremath{-}}x)$ state. The $\mathrm{C}\text{\ensuremath{-}}{\mathrm{C}}_{0},\mathrm{C}\text{\ensuremath{-}}{\mathrm{C}}_{\ensuremath{\pi}}$, and C-IC magnons become relativistic and sneak into the quantum ground state. We determine the competing boundaries among the $\mathrm{C}\text{\ensuremath{-}}{\mathrm{C}}_{0},\mathrm{C}\text{\ensuremath{-}}{\mathrm{C}}_{\ensuremath{\pi}}$, and C-IC magnons, especially the detailed dispersions of the C-IC magnons inside the canted phases, which can be mapped out by the transverse spin structure factors. As hx{h}_{x}hx​ (hz)({h}_{z})(hz​) increases further, the $\mathrm{C}\text{\ensuremath{-}}{\mathrm{C}}_{0}$ magnons always win the competition and emerge as the seeds to drive a transition from the $YX\text{\ensuremath{-}}x$ (or $YZ\text{\ensuremath{-}}x)$ to the ferromagnetic along the XXX (orZ)\text{or}Z)orZ) direction called the XXX-FM (or ZZZ-FM) phase. We show that the transition is in the 3D Ising universality class and it becomes the 3D XYXYXY transition at the two Abelian points. We evaluate these magnons' contributions to magnetization and specific heat at low temperatures which can be measured by various established experimental techniques. The nature of the finite-temperature transitions are also studied. Some analogies with quantum fluctuations generated multiple vortices and multiple landscapes in quantum spin glass are mentioned. The implications to cold-atom systems and some 4d4d4d and 5d5d5d materials with strong SOC are briefly discussed.