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Progress of Theoretical Physics

Thermal Bethe-Ansatz Method for the Spin-1/2 XXZ Heisenberg Chain

Tohru Koma

Published Apr 1, 1989• 40 citations
Publisher

Abstract

A new method is proposed for calculating the free energy of the one-dimensional spin-1/2 XXZ Heisenberg model. The partition function is written in terms of the transfer matrix for a two-dimensional Ising system, whose maximum eigenvalue is obtained by the Bethe-ansatz method leading to the free energy in the thermodynamic limit. This method uses no such assumption as the completeness of the Bethe states that has been proved only partially and yields better results than the previous methods do.

DOI
10.1143/ptp.81.783

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A new method is proposed for calculating the free energy of the one-dimensional spin-1/2 XXZ Heisenberg model. The partition function is written in terms of the transfer matrix for a two-dimensional Ising system, whose maximum eigenvalue is obtained by the Bethe-ansatz method leading to the free energy in the thermodynamic limit. This method uses no such assumption as the completeness of the Bethe states that has been proved only partially and yields better results than the previous methods do.
We review recent results on the Bethe Ansatz solutions for the eigenvalues of the transfer matrix of an integrable open XXZ quantum spin chain using functional relations which the transfer matrix obeys at roots of unity. First, we consider a case where at most two of the boundary parameters {{alpha−\\alpha_-alpha−​,alpha+\\alpha_+alpha+​,beta−\\beta_-beta−​,beta+\\beta_+beta+​}} are nonzero. A generalization of the Baxter T−QT-QT−Q equation that involves more than one independent QQQ is described. We use this solution to compute the boundary energy of the chain in the thermodynamic limit. We conclude the paper with a review of some results for the general integrable boundary terms, where all six boundary parameters are arbitrary.
The Heisenberg XXZ model is a chain model with nearest-neighbor interactions. Its thermodynamics is exactly obtained via Bethe ansatz. Recently, we developed a method to derive the high-temperature expansion of the grand potential per site of translationally invariant chain models, with periodic boundary conditions. Here we apply this approach to the XXZ model with periodic boundary conditions for the Ising limit case (t = 0) and the free fermion case (delta = 0 and h = 0), obtaining results in agreement with the literature. In this new way of obtaining the coefficients of the high-temperature expansion of the grand potential, the coefficients are derived from an auxiliary function written only in terms of open connected sub-chains.
Thermodynamics of the XXX Heisenberg model is studied. The trace of the Boltzmann weight with respect to the Hilbert space is taken in the thermodynamic limit with the number of up-spins being fixed. The expression of the trace gives an explanation why the correct thermodynamic quantities are derived from the string hypothesis. A combination of this with the previous result leads to a conclusion that the free energy can be calculated only from the Bethe ansatz equation. The method is more direct than other known methods which were used to derive the free energy.
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.