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Progress of Theoretical Physics

Finite Temperature Excitations of the XYZ Spin Chain

Masatoshi Imada

Published Jul 1, 1982• 4 citations
Publisher

Abstract

A procedure is presented to obtain finite temperature excitations of the XYZ spin chain. Numerical results agree with the analytic expressions derived by Johnson, Kinsky and McCoy at T=0. The numerical results also agree with those of the Heisenberg-Ising ring by Johnson at T ≠ 0. Sample dispersion curves are illustrated for the “scaling region”, which is expected to have and equivalence to the massive Thirring model. Temperature dependences are calculated for some types of elementary excitations.

DOI
10.1143/ptp.68.116

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A procedure is presented to obtain finite temperature excitations of the XYZ spin chain. Numerical results agree with the analytic expressions derived by Johnson, Kinsky and McCoy at T=0. The numerical results also agree with those of the Heisenberg-Ising ring by Johnson at T ≠ 0. Sample dispersion curves are illustrated for the “scaling region”, which is expected to have and equivalence to the massive Thirring model. Temperature dependences are calculated for some types of elementary excitations.
We prove the existence and discuss the uniqueness of the solution of the equations proposed by Gaudin for the free energy of the $\ensuremath{\Delta}\ensuremath{\ge}1$ Heisenberg-Ising ring. A numerical technique for solving the equations is outlined, and some results of the numerical work are displayed. Comparisons are made with the previous numerical study of Bonner and Fisher with favorable results, giving support to Gaudin's formulation. The finite-temperature excitations of the system are derived, and some sample dispersion curves are given.
Elementary excitations for the integrable spin-S antiferromagnetic Heisenberg chain with an impurity spin are investigated by using the Bethe ansatz solution. Dressed holes are introduced in order to describe the elementary excitation. The characteristic energy dependence of the spectral density for the elementary excitation at zero temperature is discussed in connection with behaviour patterns of physical quantities. We extend our calculation for the elementary excitation to the case of finite temperatures, applying the method developed by Yang and Yang for one-dimensional interacting boson systems. At finite temperatures, in the case where , the spectral density shows a divergence property at zero excitation energy irrespective of the temperature, whereas in the case where , the peak structure develops in the low-energy region as the temperature is decreased.
We consider the XYZXYZXYZ chain model of arbitrary spin SSS in the high-temperature region, with external magnetic field and single-ion anisotropy term. Our high-temperature expansion of the Helmholtz free energy is analytic in the parameters of the model for SSS, which may range from 1∕21∕21∕2 to the classical limit of infinite spin $(S\ensuremath{\rightarrow}\ensuremath{\infty})$. Our expansion is carried out up to order ${(J\ensuremath{\beta})}^{5}$. Our results agree with numerical results of the specific heat per site for S=1∕2S=1∕2S=1∕2, obtained by the Bethe ansatz, with h=0h=0h=0 and D=0D=0D=0. Finally, we show that the magnetic susceptibility and magnetization of the quantum model can be well approximated by their classical analog in this region of temperature.
We present a novel treatment of finite temperature properties of the onedimensional Hubbard model. Our approach is based on a Trotter-Suzuki mapping utilizing Shastry’s classical model and a subsequent investigation of the quantum transfer matrix. We derive non-linear integral equations for three auxiliary functions which have a clear physical interpretation of elementary excitations of spin type and charge excitations in lower and upper Hubbard bands. This allows for a transparent analytical study of certain limiting cases as well as for precise numerical investigations. We present data for the specific heat, magnetic and charge susceptibilities for various particle densities and coupling strengths U. The structure exposed by these curves is discussed in terms of the elementary charge and spin excitations. Special emphasis is placed on the study of the low-temperature behavior within our ab initio approach confirming the scaling predictions by Tomonaga-Luttinger liquid theory. In addition we make contact with the “dressed energy ” formalism established for the analysis of ground state properties.
We consider here the massive Thirring model regularized with the XYZ\mathrm{XYZ}XYZ spin chain. We numerically calculate the mass ratios of particles which lie in the discrete part of the spectrum and obtain results in accordance with the DHN formula and in disagreement with recent calculations in the literature based on the numerical Bethe ansatz and infinite momentum frame methods. We also analyze the short distance behavior of these states and evaluate the conformal dimensions. This paper, taken together with the previous one for the sine-Gordon model, confirms the duality relation between two models formulated by Klassen and Melzer [Int. J. Mod. Phys. A 8, 4131 (1993)].