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arXiv:cond-mat/9809401

Vortices in Quantum Spin Systems

John Schliemann, Franz G. Mertens

Published Sep 30, 1998• 0 citations
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Abstract

We examine spin vortices in ferromagnetic quantum Heisenberg models with planar anisotropy on two-dimensional lattices. The symmetry properties and the time evolution of vortices built up from spin-coherent states are studied in detail. Although these states show a dispersion typical for wave packets, important features of classical vortices are conserved. Moreover, the results on symmetry properties provide a construction scheme for vortex-like excitations from exact eigenstates, which have a well-controlled time evolution. Our approach works for arbitrary spin length both on triangular and square lattices.

DOI
10.48550/arxiv.cond-mat/9809401

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•Nov 10, 2013•0 citations
We examine spin vortices in ferromagnetic quantum Heisenberg models with planar anisotropy on two-dimensional lattices. The symmetry properties and the time evolution of vortices built up from spin-coherent states are studied in detail. Although these states show a dispersion typical for wave packets, important features of classical vortices are conserved. Moreover, the results on symmetry properties provide a construction scheme for vortex-like excitations from exact eigenstates, which have a well-controlled time evolution. Our approach works for arbitrary spin length both on triangular and square lattices.
Physical configurations corresponding to topological excitations present in the XY limit of a quantum spin 1/2 Heisenberg ferromagnet, are investigated on a two dimensional square lattice. Quantum vortices(anti-vortices) are constructed in terms of terms of coherent spin field components and the crucial role of the associated Wess-Zumino term is highlighted. It is shown that this term can identify a large class of vortices(anti-vortices). In particular the excitations with odd topological charge belonging to this class, are found to exhibit a self-similar pattern regarding the internal charge distribution. Our formalism is distinctly different from the coventional approach for the construction of quantum vortices(anti-vortices).
Spin wave theory plays a central role in the analysis of low-energy properties of the quantum ferromagnetic Heisenberg model. In this work, we present a first step towards a rigorous justification of this approximation in the context of quantum dynamics in one dimension. According to spin wave theory, the evolution of a spin wave state should be well approximated by its free evolution with the appropriate energy of free quasiparticles called magnons. Our main result establishes this idea rigorously for a class of spin wave packets. We focus on spin-12\frac{1}{2}21​ systems, and consider wave packets composed of two magnons. We argue our result is optimal with respect to the timescales.
The lattice of plane rotors which we have considered consists of two-dimensional spins on a two-dimensional lattice. While it is an intrinsically classical system, it shows many of the features of models which more realistically describe magnetic systems coupled in two-dimensional arrays. We start with the static properties and show how to derive generalizations of the theorems of Mermin and Wagner, and Jasnow and Fisher. The latter theorems require that the spontaneous magnetization vanish for two-dimensional Heisenberg systems. We consider more general interactions between spins than that implied by the Heisenberg model. We also note that higher moments of the magnetization must vanish as well. We derive explicit upper bounds on the mean-square magnetization when anisotropy is present. As an aid to visualization, we propose a dynamics for the lattice of plane rotors. We construct the dynamics in such a way as to leave the static statistical properties unaffected. A spin-wave-like approximation in this dynamics reduces the problem to a typical small-oscillations problem. We show (both rigorously and in the spin-wave-like approximation) that the qualitative effects of anisotropies and fields in a preferred direction are the same for certain properties.
This paper has been withdrawn and replaced by a new version entitled "Vortex quantum dynamics of two dimensional lattice bosons" By Netanel H. Lindner, Assa Auerbach, Daniel P. Arovas, posted in arXiv:0810.2604. The calculations and results of the original manuscript were correct, and lead to several important conclusions about vortex dynamics and Hall conductance. However, we have dropped our conjecture, that vortices exhibit mutual Fermi statistics. The conjecture was based on the antisymmetry under exchange of two vortices, as revealed in the lowest eight many-body eigenstates of 16 sites toroidal lattices. Appealing to an effective Harper bosons model on larger lattices, revealed vortex eigenstates with both Bose and Fermi permutation symmetries, although vortices do carry spin half quantum numbers. It may exemplify that a general non-relativistic "spin-statistics theorem" in two dimensions does not hold.
We study the localization of vortices in a three-dimensional band-gap. We report that not only planar, two-dimensional matter-wave gap vortices persist in the presence of the third dimension, but also the three-dimensional lattice may support novel types of quite unusual vortex-like structures which survive perturbations of both phase and atomic density. One remarkable result of the lattice geometry is that we may study coherent interactions of planar vortices, and even find stationary structures of an arbitrarily long stack of vortices. We find that the lattice stabilizes the vortex line, and leads to robust evolution of the perturbed state. The three-dimensional geometry also allows the existence of exotic vortex states which have no two-dimensional equivalent
We investigate the quantum dynamics of the spins on different Heisenberg antiferromagnetic spin lattice systems. Firstly, we applied the coupled-cluster method to the spin-1/2 antiferromagnetic \emph{XXZ} model on a square lattice by employing an approximation which contains two-body long-range correlations and high-order four-body local correlations. Improvement is found for the ground-state energy, sublattice magnetization, and the critical anisotropy when comparing with the approximation including the two-body correlations alone. We also obtain the full excitation spectrum which is in good agreement with the quantum Monte Carlo results and the high-order spin-wave theory.Secondly, we study the longitudinal excitations of quantum antiferromagnets on a triangular lattice by a recently proposed microscopic many-body approach based on magnon-density waves. We calculate the full longitudinal excitation spectra of the antiferromagnetic Heisenberg model for a general spin quantum number in the isotropic limit. Similar to the square lattice model, we find that, at the center of the first hexagonal Brillouin zone Γ(q=0)\Gamma(\mathbf q=0)Γ(q=0) and at the magnetic ordering wavevectors ±[Q=(4π/3,0)]\pm[\mathbf Q= (4\pi/3,0)]±[Q=(4π/3,0)], the excitation spectra become gapless in the thermodynamic limit, due to the slow, logarithmic divergence of the structure factor. However, these longitudinal modes on two-dimensional models may be considered as quasi-gapped, as any finite-size effect or small anisotropy will induce a large energy gap, when compared with the counterpart of the transverse spin-wave excitations. We have also investigated the excited states of the quasi-one-dimensional quantum antiferromagnets on hexagonal lattices, including the longitudinal modes based on the magnon-density waves. A model Hamiltonian with a uniaxial single-ion anisotropy is first studied by a spin-wave theory based on the one-boson method; the ground state thus obtained is employed for the study of the longitudinal modes. The full energy spectra of both the transverse modes (i.e., magnons) and the longitudinal modes are obtained as functions of the nearest-neighbor coupling and the anisotropy constants. We have found two longitudinal modes due to the non-collinear nature of the triangular antiferromagnetic order, similar to that of the phenomenological field theory approach by Affleck. The excitation energy gaps due to the anisotropy and the energy gaps of the longitudinal modes without anisotropy are then investigated. We then compares our results for the longitudinal energy gaps at the magnetic wavevectors with the experimental results for several antiferromagnetic compounds with both integer and non-integer spin quantum numbers, and we find good agreements after the higher-order contributions are included in our calculations.