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Physical Review Letters

Semiclassical solitons and theS=1/2 Heisenberg model

Nikos Theodorakopoulos, N. C. Bacalis

Published Nov 18, 1991• 9 citations
Publisher

Abstract

We examine the statistical mechanics of the classically integrable Ishimori-Faddeev-Haldane lattice spin model. Thermodynamic properties can be exactly described in terms of an interacting soliton gas. Our semiclassical scheme can be used to calculate the low-temperature thermodynamics of the S=1/2 Heisenberg chain, in very good agreement with recent numerical calculations based on the quantum-mechanical Bethe ansatz.

DOI
10.1103/physrevlett.67.3018

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We examine the statistical mechanics of the classically integrable Ishimori-Faddeev-Haldane lattice spin model. Thermodynamic properties can be exactly described in terms of an interacting soliton gas. Our semiclassical scheme can be used to calculate the low-temperature thermodynamics of the S=1/2 Heisenberg chain, in very good agreement with recent numerical calculations based on the quantum-mechanical Bethe ansatz.
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Integrable quantum spin chains are exactly solvable quantum mechanical models of N quantum spins, of which the Heisenberg model solved by Bethe is the prototype. In the antiferromagnetic regime, such spin chains can be regarded as lattice versions of corresponding integrable relativistic quantum field theories. For integrable spin chains, quantities of physical interest (spectrum, S matrix, etc.) can be calculated exactly by direct means, starting from the microscopic Hamiltonian, while for the corresponding field theories, such exact information has been primarily obtained by indirect means, such as the “ bootstrap ” approach and semiclassical approximations. We have been focused basically on the Heisenberg XXX and XXZ chains and their generalizations. Basic aims of this work were the study of the symmetries of these models and the computation, by means of Bethe ansatz equations, of the bulk and boundary S matrices for the closed and open chains respectively...
We show that the thermodynamic Bethe ansatz equations for one-dimensional integrable many-body systems can be reinterpreted in such a way that they only code the statistical interactions, in the sense of Haldane, between particles of identical or different momenta. Thus, the thermodynamic properties of these systems can be characterized by the generalized ideal gases recently proposed by one of us. For example, the Yang-Yang δδδ-function gas is a gas with specific statistical interactions between particles of different momenta, while the Calogero-Sutherland system provides a model for an ideal gas of particles with a fractional statistics.
We discuss the ground state of the Heisenberg spin-1/2 chain in one dimension, considering nearest neighbor interactions. In the thermodynamic limit the energy is calculated analytically in several ways. We start by applying the Bethe ansatz to calculate the exact ground state energy of the XXX Heisenberg model. Next we study small perturbations of the spin anisotropy, the parameter for the interaction in the z-direction for the XXZ model, by applying time-independent perturbation theory. In addition we expand an exact integral expression of the energy of the XXZ model around the spin anisotropy in the thermodynamic limit. The results are compared to numerical solutions, which are obtained by means of exact diagonalization up to twelve lattice sites. We find that for ten or more lattice sites the numerics agree well with the expectation from the analytical calculations. Finally the concept of integrability is discussed in the context of the Heisenberg model.