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Physical Review A

Finite-temperature excitations and thermodynamics of the Δ>~1 Heisenberg-Ising ring

James D. Johnson

Published Apr 1, 1974• 35 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.9.1743

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Chinese Physics Letters•Aug 16, 2001•3 citations
We prove the existence and discuss the uniqueness of the solution of the equations proposed by Gaudin for the free energy of the \ensuremathΔ\ensuremath≥1\ensuremath{\Delta}\ensuremath{\ge}1\ensuremathΔ\ensuremath≥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.
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.
Energy spectra and thermodynamic quantities (partition function, entropy, specific heat, susceptibility) of ferrimagnetic Heisenberg chains, made up of two spin sublattices [(1/2-S], are discussed from the exact results computed for finite rings. Extrapolation of the theoretical data versus the length of finite rings N is discussed for S ranging from 1 to (5/2 and compared with previous reported findings for the [(1/2-\ensuremath{\infty}] spin chain. In the low-temperature region, similarities between the ferrimagnetic chains [(1/2-S] and the regular ferromagnetic ones (with spin S-(1/2) are used for the determination of the limiting curves.
The thermodynamics of the one-dimensional Heisenberg-Ising model for |Δ| ≪ 1 as well as of the X-Y-Z model is reduced to a set of non-linear integral equations under some plausible assumptions. It is remarkable that the number of unknown functions involved in them becomes finite when π/ cos -1Δ is a rational number for the Heisenberg-Ising model and when Kl /ζ is a rational number for the X-Y-Z model (where coupling constants Jx, Jy and Jz are parametrized by ζ, l, and Jz as Jx = Jz cn(2ζ, l) and Jy = Jz dn(2ζ, l); 1 ≥l ≥0, Kl ≥2ζ≥0 and Kl is the complete elliptic integral of the first kind of modulus l). The validity of our theory has been confirmed by the high-temperature expansion of the free energy through the second term for a general value of Δ and through the fourth them for Δ= ½.
A simple analytic theory of thermodynamics at finite temperature for the spin-1/2 antiferromagnetic Heisenberg chain is proposed based on the picture of the particle-hole pair excitations. The dispersion relation of the particle-hole pairs is derived in the formulation of thermodynamic Bethe ansatz provided that the particles and holes have the same energy and they are excited as normal modes. It is shown that the behaviour of the specific heat is in excellent agreement with the numerical and experimental results.