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On Shortening of the Classical Spin Length in OneDimensional Heisenberg Magnetic with Spin Value

F. K. Rahimov, Kh. O. Abdulloev, V. K. Fedyanin

Published Jan 1, 2000• 0 citations
OpenAlex

Abstract

The present work analyses the main models of the considered magnetic systems. By employing the method of generalized coherent states in SU(3)/SU(2) U(1) and SU(2S+1)/SU(2S) U(1) space, it is shown that the effect of shortening of the classical spin length due to quadrupole and octupole interactions occurs in magnetics with spin values S=1 and S=3/2.

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The present work analyses the main models of the considered magnetic systems. By employing the method of generalized coherent states in SU(3)/SU(2) U(1) and SU(2S+1)/SU(2S) U(1) space, it is shown that the effect of shortening of the classical spin length due to quadrupole and octupole interactions occurs in magnetics with spin values S=1 and S=3/2.
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.
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.
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.