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arXiv:1601.05067 · Physical review. B./Physical review. B

Classification of magnons in rotated ferromagnetic Heisenberg model and their competing responses in transverse fields

Fadi Sun, Jinwu Ye, Wu-Ming Liu

Published Jul 7, 2016• 9 citations
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Abstract

In this paper, we study the rotated ferromagnetic Heisenberg model (RFHM) in two different transverse fields, ${h}_{x}$ and ${h}_{z}$, which can be intuitively visualized as studying spin-orbit coupling (SOC) effects in two-dimensional (2D) Ising or anisotropic $XY$ 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 ${h}_{y}$ 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 ${h}_{x}$ $({h}_{z})$ 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 ${h}_{x}$ $({h}_{z})$ 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 $X$ ($\text{or}Z)$ direction called the $X$-FM (or $Z$-FM) phase. We show that the transition is in the 3D Ising universality class and it becomes the 3D $XY$ 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 $4d$ and $5d$ materials with strong SOC are briefly discussed.

DOI
10.1103/physrevb.94.024409

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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\ensuremath-xY\text{\ensuremath{-}}xY\ensuremath-x state. It supports not only the commensurate magnons (called C\ensuremath-C0\mathrm{C}\text{\ensuremath{-}}{\mathrm{C}}_{0}C\ensuremath-C0​ and C\ensuremath-C\ensuremathπ)\mathrm{C}\text{\ensuremath{-}}{\mathrm{C}}_{\ensuremath{\pi}})C\ensuremath-C\ensuremathπ​), 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\ensuremath-xY\text{\ensuremath{-}}xY\ensuremath-x state to a canted coplanar YX\ensuremath-xYX\text{\ensuremath{-}}xYX\ensuremath-x (YZ\ensuremath-x)(YZ\text{\ensuremath{-}}x)(YZ\ensuremath-x) state. The C\ensuremath-C0,C\ensuremath-C\ensuremathπ\mathrm{C}\text{\ensuremath{-}}{\mathrm{C}}_{0},\mathrm{C}\text{\ensuremath{-}}{\mathrm{C}}_{\ensuremath{\pi}}C\ensuremath-C0​,C\ensuremath-C\ensuremathπ​, and C-IC magnons become relativistic and sneak into the quantum ground state. We determine the competing boundaries among the C\ensuremath-C0,C\ensuremath-C\ensuremathπ\mathrm{C}\text{\ensuremath{-}}{\mathrm{C}}_{0},\mathrm{C}\text{\ensuremath{-}}{\mathrm{C}}_{\ensuremath{\pi}}C\ensuremath-C0​,C\ensuremath-C\ensuremathπ​, 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 C\ensuremath-C0\mathrm{C}\text{\ensuremath{-}}{\mathrm{C}}_{0}C\ensuremath-C0​ magnons always win the competition and emerge as the seeds to drive a transition from the YX\ensuremath-xYX\text{\ensuremath{-}}xYX\ensuremath-x (or YZ\ensuremath-x)YZ\text{\ensuremath{-}}x)YZ\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.
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 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.
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.
We use the spin functional renormalization group recently developed by two of us [Krieg and Kopietz, Phys. Rev. B 99, 060403(R) (2019)] to calculate the magnetization M(H,T)M(H,T)M(H,T) and the damping of magnons due to classical longitudinal fluctuations of quantum Heisenberg ferromagnets. In order to guarantee that for vanishing magnetic field $H\ensuremath{\rightarrow}0$, the magnon spectrum is gapless when the spin rotational invariance is spontaneously broken, we use a Ward identity to express the magnon self-energy in terms of the magnetization. In two dimensions, our approach correctly predicts the absence of long-range magnetic order for H=0H=0H=0 at finite temperature TTT. The magnon spectrum then exhibits a gap from which we obtain the transverse correlation length. We also calculate the wave-function renormalization factor of the magnons. As a mathematical byproduct, we derive a recursive form of the generalized Wick theorem for spin operators in frequency space which facilitates the calculation of arbitrary time-ordered connected correlation functions of an isolated spin in a magnetic field.
The ground state phase diagram of the alternating spin-1/2 chains with anisotropic ferromagnetic coupling under the influence of a symmetry breaking transverse magnetic field is studied. We have used the exact diagonalization technique. In the limit where the antiferromagnetic coupling is dominant, we have identified two Ising-type quantum phase transitions. We have calculated two critical fields hc1h_{c_{1}}hc1​​ and hc2h_{c_{2}}hc2​​, corresponding to the transition between different magnetic phases of the system. It is found that the intermediate state ($h_{c_{1}}
In this thesis we investigate the thermodynamic and dynamic properties of the D-dimensional quantum Heisenberg ferromagnet within the spin functional renormalization group (FRG); a formalism describing the evolution of the system’s observables as the magnetic exchange inter-action is artificially deformed. Following an introduction providing a self contained summary of the conceptual and mathematical background, we present the spin FRG as developed by Krieg and Kopietz in references [1] and [2] in chapter two. Thereto, the generating functional of the imaginary time-spin correlation functions and its exact flow equation describing the deformation process of the exchange interaction are introduced. In addition, it is highlighted that - in contrast to conventional field-theoretic FRG approaches - the related Legendre trans-formed functional cannot be defined if the exchange interaction is initially switched off. Next, we show that this limitation can be circumvented within an alternativ hybrid approach, which treats transverse and longitudinal spin fluctuations differently. The relevant functionals are introduced and the relations of the corresponding functional Taylor coefficients with the spin correlation functions are discussed. Lastly, the associated flow equations are derived and the possibility of explicit or spontaneous symmetry breaking is taken into account. In chapter three, we benchmark the hybrid formalism against a calculation of the thermo-dynamic properties of the one and two-dimensional Heisenberg model at low temperatures T and finite magnetic field H. For this purpose, we devise an anisotropic deformation scheme of the exchange interaction which allows for a controlled truncation of the infinite hierarchy of FRG flow equations. Thereby, contact with mean-field and spin-wave theory is made and the violation of the Mermin-Wagner theorem is discussed. To fulfill the latter, the truncation scheme is then complemented by a Ward identity relating the transverse self-energy and the magnetization. The resulting magnetization M (H, T ) and isothermal susceptibility χ(H, T ) are in quantitative agreement with the literature and the established behavior of the transverse correlation length and the zero-field susceptibility close to the critical point is qualitatively reproduced in the limit H → 0. Finally, we investigate the longitudinal dynamics at low temperatures. To this end, the hierarchy of flow equations is solved within the same anisotropic deformation scheme complemented by an expansion in the inverse interaction range, and the resulting longitudinal dynamic structure factor is calculated within a low-momentum expansion. In D = 3, the large phase space accessible for the decay into transverse magnons yields only a broad hump centered at zero frequency whose width scales linearly in momentum. In contrast, at low temperatures and in a certain range of magnetic fields, a well-defined quasiparticle peak with linear dispersion emerges in D ≤ 2, which we identify as zero-magnon sound. Sound velocity and damping are discussed as a function of temperature and magnetic field, and the relevant momentum-frequency window is estimated and compared to the hydrodynamic second-magnon regime.