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arXiv:cond-mat/0412698 · Journal of the Physical Society of Japan

Nonlinear Integral Equations for Thermodynamics of the \(U_{q}(\widehat{sl(r+1)})\) Perk–Schultz Model

Zengo Tsuboi, Minoru Takahashi

Published Mar 1, 2005• 9 citations
arXivHTMLPDF

Abstract

We propose a system of nonlinear integral equations (NLIE) which describes the thermodynamics of the U_{q}(\hat{sl(r+1)}) Perk-Schultz model. These NLIE correspond to a trigonometric analogue of our previous result (cond-mat/0212280), and contain only r unknown functions. In particular, they reduce to Takahashi's NLIE for the XXZ spin chain (cond-mat/0010486) if r=1. We also calculate the high temperature expansion of the free energy. In particular for r=1 case, we have succeeded to derive the coefficients of order O((\frac{J}{T})^{99}).

DOI
10.1143/jpsj.74.898

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The quantum mechanical analogue of the classical Perk - Schultz model is considered which comprises the Uimin - Sutherland model and the integrable t - J chain. The quantum transfer matrix of these systems is established and the eigenvalue equations are obtained by an algebraic Bethe ansatz. Only the largest eigenvalue is needed for the calculation of the free energy of the quantum chain at finite temperature. The Bethe ansatz equations for the leading eigenvalue are transformed into a set of integral equations for some appropriately defined auxiliary functions. Furthermore, the eigenvalue of the quantum transfer matrix is expressed in terms of these functions. The integral formulation allows for taking the limit of infinite Trotter - Suzuki number analytically. The low-temperature limit of the free energy is obtained analytically and for intermediate temperatures numerical results are presented.
Abstract A new nonlinear integral equation (NLIE) describing the thermodynamics of the Heisenberg spin chain is derived based on the t − W relation of the quantum transfer matrices. The free energy of the system in a magnetic field is thus obtained by solving the NLIE. This method can be generalized to other lattice quantum integrable models. Taking the SU(3)-invariant quantum spin chain as an example, we construct the corre- sponding NLIEs and compute the free energy. The present results coincide exactly with those obtained via other methods previously.