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Journal of the Physical Society of Japan

Correlation Length for the XXZ Spin Chain at Finite Temperature

Hideyuki Mizuta, Taro Nagao, Miki Wadati

Published Nov 15, 1994• 17 citations
Publisher

Abstract

Baxter's formula is extended to treat the thermodynamic properties of quantum spin chains. As an application of such formulation which we call finite temperature Baxter's formula, the XXZ Heisenberg spin chain is investigated. Combination of the formula and the finite size expansion provides a systematic method to evaluate low-temperature expansions of the thermodynamic quantities. The correlation length for the XXX spin chain is explicitly evaluated.

DOI
10.1143/jpsj.63.3951

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Baxter's formula is extended to treat the thermodynamic properties of quantum spin chains. As an application of such formulation which we call finite temperature Baxter's formula, the XXZ Heisenberg spin chain is investigated. Combination of the formula and the finite size expansion provides a systematic method to evaluate low-temperature expansions of the thermodynamic quantities. The correlation length for the XXX spin chain is explicitly evaluated.
Thermodynamic quantities of solvable models such as one–dimensional Heisenberg model can be obtained through the Finite temperature Baxter formula. For the low–temperature expansions, the estimate of the finite size corrections by use of the Euler–Maclaurin formula is essential. There has been a question on the convergence of such calculations. In this paper, we confirm that the Euler–Maclaurin formula gives the correct low–temperature expansion of the free energy for the XXZ Heisenberg model.
We investigate the thermal properties of s=1/2 and 1 antiferromagnetic Heisenberg spin chains by an expansion in finite lattices. The expansion is based on expressing the quantity of interest, in the thermodynamic limit, as a sum over contributions from different length scales. The first few terms in the expansion, obtained by diagonalizing chains of length up to 13 for s=1/2 and length up to 9 for s=1, provide an accurate estimate for the internal energy and specific heat for T/J\ensuremath{\ge}0.25 and 0.35, respectively. At lower temperatures, by studying the exponential decay of the various terms in the expansion, we estimate the correlation length \ensuremath{\xi}. We get good estimates for \ensuremath{\xi} down to T/J=0.06 for s=1/2 and T/J=0.02 for s=1. We find that for s=1 \ensuremath{\xi} increases much more slowly than for s=1/2 and is consistent with a value of \ensuremath{\xi}\ensuremath{\approxeq}7 at T=0.
We communicate results on correlation functions for the spin-1/2 Heisenberg chain in two particularly important cases: (a) for the infinite chain at arbitrary finite temperature T, and (b) for finite chains of arbitrary length L in the ground state. In both cases we present explicit formulas expressing the short-range correlators in a range of up to seven lattice sites in terms of a single function ω encoding the dependence of the correlators on T (L). These formulas allow us to obtain accurate numerical values for the correlators and derived quantities like the entanglement entropy. By calculating the low T (large L) asymptotics of ω we show that the asymptotics of the static correlation functions at any finite distance are T(2) (1/L(2)) terms. We obtain exact and explicit formulas for the coefficients for up to eight sites.
Abstract This chapter presents the extension of the Bethe ansatz to finite temperature, the thermodynamic Bethe ansatz, for the antiferromagnetic isotropic Heisenberg quantum spin chain, the XXX quantum spin chain. It discusses how the added complications of this model arise from the more complicated structure of excitations of the quantum spin chain, the complex string excitations, which have to be included in the Bethe ansatz thermodynamics. It derives the integral equations of the thermodynamic Bethe ansatz for the XXX quantum spin chain and mentions explicit formulas for the free energy of the quantum spin chain and some interesting physical quantities, especially making contact with predictions of conformal symmetry.