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Physica Scripta

Generalized Coherent States and the Continuous Heisenberg XYZ Model With One-Ion Anisotropy

Vladmir G. Makhankov, R. Myrzakulov, Алексей Владимирович Маханьков

Published Mar 1, 1987• 17 citations
Publisher

Abstract

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.

DOI
10.1088/0031-8949/35/3/003

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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.
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.
This paper develops a semiclassical framework for one-dimensional Heisenberg magnets with spin S>1/2, where classical approaches are reliable only as S→∞ and can miss quantum contributions from multipole moments. The aim is to construct a workable mathematical apparatus that connects the quantum lattice Hamiltonian to an effective classical field description while incorporating an effective reduction of the classical spin length. The analysis starts from an anisotropic Heisenberg chain with nearest-neighbour exchange and single-ion anisotropy. After outlining the continuum (long-wavelength) reduction, including factorization of slowly varying fields and replacement of commutators by Poisson brackets, the semiclassical limit is formulated using a Hartree product ansatz and generalized SU(N) coherent states for each lattice site. For spin S=1 the coherent state is built on the SU(3)/[SU(2)×U(1)] manifold. Two Euler angles define the orientation of the classical spin vector, an additional angle describes rotation of the quadrupole tensor about this vector, and a real parameter g controls the redistribution between dipolar and quadrupolar sectors. Averaging the spin operators yields explicit classical components and shows that the standard constraint |S|^2=1 is violated. Instead, an identity relates the reduced spin length to a combination of double correlators, demonstrating that quadrupolar degrees of freedom quantitatively produce a measurable renormalization of the effective classical spin. For spin S=3/2 the construction is generalized to SU(4) coherent states, where both quadrupole and octupole moments arise naturally. The averaged spin operators again violate spin-length conservation and, moreover, simple projection sum rules. The corresponding SU(4) identities involve combinations of triple correlators and introduce parameters g and k that quantify reductions due to quadrupolar and octupolar sectors, respectively; the SU(3) case is recovered when k=0. The framework provides a transparent route to include multipolar physics in semiclassical dynamics and is relevant for interpreting nonlinear collective excitations, including soliton-like modes, in quantum spin chains.
We introduce a classical limit of the dynamics of quantum spin systems based on coherent states of SU(N)\mathrm{SU}(N)SU(N), where NNN is the dimension of the local Hilbert space. This approach, which generalizes the well-known Landau-Lifshitz dynamics from SU(2) to SU(N)\mathrm{SU}(N)SU(N), provides a better approximation to the exact quantum dynamics for a large class of realistic spin Hamiltonians, including $S\ensuremath{\ge}1$ systems with large single-ion anisotropy and weakly coupled multispin units, such as dimers or trimers. We illustrate this idea by comparing the spin structure factors of a single-ion S=1S=1S=1 model that are obtained with the SU(2) and SU(3) classical spin dynamics against the exact solution.
The topic of this Bachelor's thesis is the quantum-to-classical crossover using generalized coherent states (GCS). In other words, the thesis consists of a mathematically rigorous consideration of the transition between quantum theory and classical theory, and how this can be carried out using the theory of GCS. The necessary mathematical background for this is also reviewed. For the theory of generalized coherent states, Lie algebras and Lie groups are first reviewed. Lie groups are groups with a structure of a differentiable manifold. For the Lie groups we consider, there exist coresponding Lie algebras, whose elements generate the Lie groups using the exponential map. Generalized coherent states are a generalization of Glauber states, which are continuous quantum states used in describing the harmonic oscillator. The generalization is carried through using some Lie group other than the Heisenberg-Weyl group used in defining the Glauber states. A special property of GCS is their semiclassical structure: a structure resembling the classical phase space may be defined in the differentiable manifold used in generating the GCS. An example of GCS are the spin coherent states, generated using the Lie group SU(2). For the coherent states to be applicable in the quantum-to-classical crossover, one has to also find some symmetry so that the system under consideration may be described using a single GCS. In the reference literature it is assumed that a global symmetry enables the description of a system of N elements by a coherent state corresponding to a coherent state of a single element. In the spin chain example however, the symmetry can be explicitly applied on the system and it turns out that this assumption is valid: using suitable symmetries, a spin chain may be represented by a single spin coherent state. The quantum-to-classical crossover can be carried out by defining some classical operators whose expectation value with respect to the GCS corresponds to the functions of classical mechanics. In addition to this, the coherent states in question have to fulfill some requirements. In the case of spin coherent states these requirements are fulfilled and the quantum-to-classical crossover may be carried through.