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Direct Calculation of Thermodynamic Quantities for Heisenberg Model

Go Kato, Miki Wadati

Published Jan 1, 2002• 12 citations
OpenAlex

Abstract

The XXX Heisenberg model is studied at finite temperature. The free energy is derived without recourse to Thermal Bethe Ansatz method and Quantum Transfer Matrix method. The result perfectly agrees with the free energy derived by Thermal Bethe Ansatz method. An explicit expression of the cluster expansion coefficient in arbitrary order is presented for the first time.

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The XXX Heisenberg model is studied at finite temperature. The free energy is derived without recourse to Thermal Bethe Ansatz method and Quantum Transfer Matrix method. The result perfectly agrees with the free energy derived by Thermal Bethe Ansatz method. An explicit expression of the cluster expansion coefficient in arbitrary order is presented for the first time.
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.
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.
The density matrix renormalization group (DMRG) method is applied to the anisotropic Heisenberg chain at finite temperatures. The free energy of the system is obtained using the quantum transfer matrix which is iteratively enlarged in the imaginary time direction. The magnetic susceptibility and the specific heat are calculated down to T=0.01J and compared with the Bethe ansatz results. The agreement including the logarithmic correction in the magnetic susceptibility at the isotropic point is fairly good.
Thermodynamic properties of the quantum Heisenberg spin chains with S=1/2,S=1/2,S=1/2, 1, and 3/23/23/2 are investigated using the transfer-matrix renormalization-group method. The temperature dependence of the magnetization, susceptibility, specific heat, spin-spin correlation length, and several other physical quantities in a zero or finite applied field are calculated and compared. Our data agree well with the Bethe ansatz, exact diagonalization, and quantum Monte Carlo results and provide further insight into the quantum effects in the antiferromagnetic Heisenberg spin chains.