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physica status solidi (b)

Time‐dependent solitary solutions of the quantum‐mechanical anisotropic Heisenberg chain

A. Freudenhammer

Published Mar 1, 1986• 1 citation
Publisher

Abstract

Abstract A solitary time‐dependent solution for a linear chain of quantum‐mechanical Heisenberg spins is given. The interaction is of the anisotropic exchange type. An approximation to a one‐spin‐flip of the Hamiltonian on the basis of local spin quantization directions leads to the result that kink solutions only exist for a static kink.

DOI
10.1002/pssb.2221340120

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Abstract A solitary time‐dependent solution for a linear chain of quantum‐mechanical Heisenberg spins is given. The interaction is of the anisotropic exchange type. An approximation to a one‐spin‐flip of the Hamiltonian on the basis of local spin quantization directions leads to the result that kink solutions only exist for a static kink.
Abstract This chapter introduces the Heisenberg model, a fully quantum mechanical model that describes the magnetism of localized magnetic moments. The one-dimensional version of the Heisenberg model, the Heisenberg quantum spin chain, provides a good picture of magnetic materials that belong to a class of insulating magnetic materials where the interaction of the magnetic moments in one particular direction is much larger than in the perpendicular directions, and which can be described with high accuracy as quasi- one-dimensional magnets. A detailed description of the Heisenberg quantum spin chain is followed by a discussion of its various special cases, in particular the special case of the anisotropic Heisenberg quantum spin chain, the so-called XXZ quantum spin chain. It considers the solution of eigenvalue problem of this quantum spin and leads to Bethe’s conjecture for the wave function.
We investigate whether quantum spin rings with nearest-neighbor Heisenberg or Ising exchange interactions can host solitary states. Using complete diagonalization techniques the system is described without classical or semiclassical approximation. In this case definitions used in connection with classical solitons are not applicable, one needs to redefine what solitary objects on a quantum spin system with translational symmetry ought to be. Thus, we start our contribution by defining which quantum states possess solitary character. In addition we discuss useful observables in order to visualize solitary quantum states. Then we demonstrate for various quantum spin rings that solitary quantum states indeed exist, and that they are moving around the spin ring without changing their shape in the course of time.
In the recent literature, there have been discussions of nonlinear spin excitations in one-dimensional, anisotropic Heisenberg spin chains. These have the character of localized excitations, which emerge from the time-dependent classical equations of motion. We show that for a given frequency, in a finite spin chain one has a hierarchy of nonlinear excitations, whose envelopes have the appearance of a one soliton, two soliton,..., states. Also, we consider the nature of these nonlinear excitations in an applied magnetic field of frequency \ensuremath{\omega}, and explore aspects of the transient behavior of the system in response to a time varying external field. \textcopyright{} 1996 The American Physical Society.
The spin evolution equation of an isotropic, quantum ferromagnetic Heisenberg chain is analyzed in the continuum approximation using spin-coherent states. The advantages of this approach are discussed. Magnetic solitary-wave solutions are found, and the expectation values of the energy, momentum, and angular momentum corresponding to these solutions are determined. The energy-momentum dispersion relation for the nonlinear excitations is derived. The semiclassical spectrum is shown to arise when quantum effects are neglected by using a random-phase approximation to calculate certain expectation values. On including the quantum effects, it is found that the spectrum comprises two branches: a lower-energy branch of spin-wave-like, small-amplitude solitary waves with small quantum corrections, and a higher branch of particlelike large-amplitude solitary waves subject to significant quantum corrections for small S values. A heuristic discussion of the stability of these excitations is presented. A physical interpretation of the dispersion relation obtained is given.
From the point of view of the quantum lattice liquid, an attempt is made to understand the nonlinear solitonlike excitations in the quantum anisotropic Heisenberg chain with the help of the coherent-state method. It is argued that the modified terms of the nonlinear Schr\"odinger equation are strongly restricted by the relation between the continuum approximation and the semiclassical approximation represented by two smallness parameters \ensuremath{\eta} and \ensuremath{\epsilon}, respectively. When assuming that \ensuremath{\eta} and \ensuremath{\epsilon} are of the same order (\ensuremath{\eta}\ensuremath{\sim}\ensuremath{\epsilon}), after retaining terms of order O(${\mathrm{\ensuremath{\epsilon}}}^{6}$), the equation of motion of the Bose operators satisfies the cubic nonlinear Schr\"odinger equation. Envelope-kink excitations and energy gaps are obtained. Some relevant physical results are also discussed.