On the Reduction of Classical Spin in One-dimensional Heisenberg Magnets with Spin S>1/2

Abstract

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.