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Physica Scripta

Application of spin coherent states to 1-dHeisenberg model

D. Kapor, M. J. Škrinjaŕ, S. D. Stojanović

Published Apr 1, 1989• 3 citations
Publisher

Abstract

Method of spin coherent states has been recently extensively applied to different Heisenberg models in order to formulate a correct procedure of the reduction of quantum models to continual Landau-Lifshitz model. This paper deals with the applicability of the method from the point of view of the time evolution of spin coherent states.
DOI
10.1088/0031-8949/39/4/018

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Generalized Coherent States (GCS) are constructed (and discussed) in order to study quasiclassical behaviour of quantum spin models of the Heisenberg type. Several such models are taken to their semiclassical limits, whose form depends on the spin value as well as the Hamiltonian symmetry. In the continuum approximation, SU(2)/U(1) GCS when applied give rise to the well known Landau - Lifshitz classical phenomenology. For arbitrary spin values one obtains a lattice of coupled nonlinear oscillators. Corresponding classical continuum models are described as well.
We use spin-coherent states as the basis of a time-dependent variational ansatz for a semiclassical description of a large family of Heisenberg models. In addition to following common approaches we also evaluate the square variance of the Hamiltonian in terms of coherent states. This quantity turns out to have a natural interpretation with respect to time-dependent solutions of the equations of motion and allows an estimate of quantum fluctuations in a semiclassical regime to be made. The general results are applied to solitons, instantons and vortices in several one- and two-dimensional models.
The coherent-anomaly method (CAM) is applied to the Heisenberg model to test the applicability of the CAM for quantum spin systems. The Weiss, Bethe and constant coupling approximations are tried for the Heisenberg model on the simple cubic lattice and estimate the critical exponents of the susceptibility and spontaneous magnetization using the CAM. The results show that the CAM is also powerful for quantum spin systems. The detailed results of the Bethe approximation of the spin-1/2 isotropic Heisenberg model are presented.
On the basis of the method of generalized coherent states (GCS) of the SU(2) group we formulate the reduction procedure of the quantum lattice Heisenberg XYZ model to the continuum Landau-Lifshitz model. The choice of the GCS representation is determined (i) by their proximity to the corresponding classical states and (ii) by similarity of the geometric structure of the homogeneous spaces on which the SU(2) GCS and the vector of the magnetic moment of the corresponding Landau-Lifshitz model are defined. The present reduction procedure is allowed only in the case of weak anisotropy. Accounting for a weak one-ion anisotropy leads to the Landau-Lifshitz model with a renormalized anisotropy tensor. The contribution of such an anisotropy vanishes for the s = 1/2 case.
We solve the operator ordering problem for the quantum continuous integrable su(1,1) Landau–Lifshitz model, and give a prescription to obtain the quantum trace identities, and the spectrum for the higher-order local charges. We also show that this method, based on operator regularization and renormalization, which guarantees quantum integrability, as well as the construction of self-adjoint extensions, can be used as an alternative to the discretization procedure, and unlike the latter, is based only on integrable representations.