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In-commensurate Skyrmion crystals, devil staircases and multi-fractals due to spin orbit coupling in a lattice system

Fadi Sun, Jinwu Ye

Published Mar 1, 2016• 1 citation
OpenAlex

Abstract

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.

Related papers

Two classes of organization principle: quantum/topological phase transitions meet complete/in-complete devil staircases and their experimental realizations

Fadi Sun, Jinwu Ye

There exists many quantum or topological phases in Nature. One well known organization principle is through various quantum or topological phases transitions between or among these phases. Another is through either complete or in-complete devil staircases in their quantized forms. Here, we show that both classes of organization principle appear in an experimentally accessible system: strongly interacting spinor bosons subject to any of the linear combinations of the Rashba and Dresselhaus spin-orbit coupling (SOC) in the space of the two SOC parameters $ ( α, β) $ in a square lattice. In the strong coupling limit, it leads to a new quantum spin model called Rotated Ferromagnetic Heisenberg model (RFHM). The RFHM leads to rich and unconventional magnetic phases even in a bipartite lattice. For the first class, by identifying a suitable low energy mode, we investigate a new quantum Lifshitz transition from the Y-x to the IC-SkX-y phase. For the second class, we introduce the topological rational and irrational winding numbers $ W $ to characterize the incomplete or complete devil staircases and also perform their quantizations. The IC-YZ-x/LQx phases form a Cantor set with a fractal dimension along the complete devil staircase. They also take most of measures in the incomplete devil staircases when $ β\ll α$. Quantum chaos and quantum information scramblings along the diagonal line $ α=β$ are discussed. Implications on un-conventional magnetic ordered phases detected in the 4d- or 5d-orbital strongly correlated materials with SOC and in the current or near future cold atom systems are presented.

arXiv:1603.00451•Mar 1, 2016•2 citations

Quantum incommensurate skyrmion crystals and commensurate to in-commensurate transitions in cold atoms and materials with spin–orbit couplings in a Zeeman field

Fadi Sun, Jinwu Ye, Wu‐Ming Liu

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.

arXiv:1502.05338•New Journal of Physics•Aug 16, 2017•13 citations

Quantum Lifshitz transitions generated by order from quantum disorder in strongly correlated Rashba spin-orbital coupled systems

Fadi Sun, Jinwu Ye

We study the system of strongly interacting spinor bosons in a square lattice subject to the isotropic Rashba SOC $ α=β$. It supports collinear spin-bond correlated magnetic Y-x phase, a gapped in-commensurate (IC-) co-planar IC-XY-y phase, a non-coplanar commensurate (C-) $ 3 \times 3 $ Skyrmion crystal phase (SkX). The state at the Abelian point $ α=β=π/2 $ is just an AFM state in a rotated basis. Slightly away from the point, we identify a spurious $ U(1) $ symmetry, develop a novel and non-perturbative method to calculate not only the gap, but also the excitation spectrum due to the order from quantum disorder (OFQD) mechanism. We construct a symmetry based effective action to investigate the quantum Lifshitz transition from the Y-x state to the IC-XY-y state and establish the connection between the phenomenological parameters in the effective action and those evaluated by the microscopic non-perturbative OFQD analysis in the large $ S $ limits. Experimental implications on cold atoms and some 4d or 5d Kitaev materials are discussed.

arXiv:2202.12068•Feb 24, 2022•0 citations

Quantum Lifshitz transitions generated by order from quantum disorder in strongly correlated Rashba spin-orbit-coupled systems

Fadi Sun, Jinwu Ye

We study the system of strongly interacting spinor bosons in a square lattice subject to the isotropic Rashba spin-orbit coupling $\ensuremath{\alpha}=\ensuremath{\beta}$. It supports a collinear spin-bond correlated magnetic Y-x phase, a gapped coplanar in-commensurate (IC) XY-y phase, and a noncoplanar commensurate (C) $3\ifmmode\times\else\texttimes\fi{}3$ Skyrmion crystal phase. The state at the Abelian point $\ensuremath{\alpha}=\ensuremath{\beta}=\ensuremath{\pi}/2$ is just a ferromagnetic state in a rotated basis. Slightly away from the point, we identify a spurious $\text{U}(1)$ symmetry, develop a nonperturbative method to calculate not only the gap but also the excitation spectrum due to the order from quantum disorder (OFQD) mechanism. We construct a symmetry-based effective action to investigate the quantum Lifshitz transition from the Y-x state to the IC XY-y state and establish the connection between the phenomenological parameters in the effective action and those evaluated by the microscopic nonperturbative OFQD analysis in the large $S$ limits. Experimental implications on cold atoms and some $4d$ or $5d$ Kitaev materials are discussed.

Physical Review A•Sep 29, 2021•2 citations

Fermionic Hubbard model with Rashba or Dresselhaus spin–orbit coupling

Fadi Sun, Jinwu Ye, Wu-Ming Liu

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.

arXiv:1601.01642•New Journal of Physics•Apr 20, 2017•14 citations

Rotated Heisenberg model in a Zeeman field

Fadi Sun, Jinwu Ye, Wu‐Ming Liu

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 ($z_x = 1$, $z_y = 3$) 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.

•Feb 18, 2015•0 citations
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.
There exists many quantum or topological phases in Nature. One well known organization principle is through various quantum or topological phases transitions between or among these phases. Another is through either complete or in-complete devil staircases in their quantized forms. Here, we show that both classes of organization principle appear in an experimentally accessible system: strongly interacting spinor bosons subject to any of the linear combinations of the Rashba and Dresselhaus spin-orbit coupling (SOC) in the space of the two SOC parameters $ ( α, β) $ in a square lattice. In the strong coupling limit, it leads to a new quantum spin model called Rotated Ferromagnetic Heisenberg model (RFHM). The RFHM leads to rich and unconventional magnetic phases even in a bipartite lattice. For the first class, by identifying a suitable low energy mode, we investigate a new quantum Lifshitz transition from the Y-x to the IC-SkX-y phase. For the second class, we introduce the topological rational and irrational winding numbers $ W $ to characterize the incomplete or complete devil staircases and also perform their quantizations. The IC-YZ-x/LQx phases form a Cantor set with a fractal dimension along the complete devil staircase. They also take most of measures in the incomplete devil staircases when $ β\ll α$. Quantum chaos and quantum information scramblings along the diagonal line $ α=β$ are discussed. Implications on un-conventional magnetic ordered phases detected in the 4d- or 5d-orbital strongly correlated materials with SOC and in the current or near future cold atom systems are presented.
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 the system of strongly interacting spinor bosons in a square lattice subject to the isotropic Rashba SOC $ α=β$. It supports collinear spin-bond correlated magnetic Y-x phase, a gapped in-commensurate (IC-) co-planar IC-XY-y phase, a non-coplanar commensurate (C-) $ 3 \times 3 $ Skyrmion crystal phase (SkX). The state at the Abelian point $ α=β=π/2 $ is just an AFM state in a rotated basis. Slightly away from the point, we identify a spurious $ U(1) $ symmetry, develop a novel and non-perturbative method to calculate not only the gap, but also the excitation spectrum due to the order from quantum disorder (OFQD) mechanism. We construct a symmetry based effective action to investigate the quantum Lifshitz transition from the Y-x state to the IC-XY-y state and establish the connection between the phenomenological parameters in the effective action and those evaluated by the microscopic non-perturbative OFQD analysis in the large $ S $ limits. Experimental implications on cold atoms and some 4d or 5d Kitaev materials are discussed.
We study the system of strongly interacting spinor bosons in a square lattice subject to the isotropic Rashba spin-orbit coupling $\ensuremath{\alpha}=\ensuremath{\beta}$. It supports a collinear spin-bond correlated magnetic Y-x phase, a gapped coplanar in-commensurate (IC) XY-y phase, and a noncoplanar commensurate (C) $3\ifmmode\times\else\texttimes\fi{}3$ Skyrmion crystal phase. The state at the Abelian point $\ensuremath{\alpha}=\ensuremath{\beta}=\ensuremath{\pi}/2$ is just a ferromagnetic state in a rotated basis. Slightly away from the point, we identify a spurious U(1)\text{U}(1)U(1) symmetry, develop a nonperturbative method to calculate not only the gap but also the excitation spectrum due to the order from quantum disorder (OFQD) mechanism. We construct a symmetry-based effective action to investigate the quantum Lifshitz transition from the Y-x state to the IC XY-y state and establish the connection between the phenomenological parameters in the effective action and those evaluated by the microscopic nonperturbative OFQD analysis in the large SSS limits. Experimental implications on cold atoms and some 4d4d4d or 5d5d5d Kitaev materials are discussed.
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 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.