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arXiv:cond-mat/0510441 · Physical Review B

High-temperature series expansion of the Helmholtz free energy of the quantum spin- S XYZ chain

Onofre Rojas, S. M. de Souza, E. V. Corrêa Silva, M. T. Thomaz

Published Nov 22, 2005• 6 citations
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Abstract

We consider the $XYZ$ chain model of arbitrary spin $S$ 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 $S$, which may range from $1∕2$ 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 $S=1∕2$, obtained by the Bethe ansatz, with $h=0$ and $D=0$. 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.

DOI
10.1103/physrevb.72.172414

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We report analytical calculations of the Helmholtz free energy of non-integrable anisotropic quantum XXZ chains in the high-temperature regime for several values of the spin S . Single-ion anisotropy and interaction with an external magnetic field are taken into account. The seven lowest-order terms in the high-temperature expansion of Helmholtz free energy are obtained. Our results contribute to the existing literature on high-temperature expansions and numerical studies of those models by discussing the effects of anisotropy upon their high-temperature thermodynamic properties, such as the average energy per site, the specific heat and magnetic susceptibility.
Recently Rojas et al. [J. Math. Phys. 43, 1390 (2002)] obtained a closed analytical expression for the coefficients of arbitrary order in the cumulant expansion of a one-dimensional periodic chain model with nearest-neighbor interaction and spatial translation invariance; that approach can be applied equally well to both nonintegrable and exact integrable models. Here, we obtain the exact analytic expressions for the six lowest-order terms of the high-temperature expansion of the Helmholtz free energy per site of the nonintegrable one-dimensional spin-111 XXZ\mathrm{XXZ}XXZ Heisenberg model. Our analytical results for the specific heat and the static magnetic susceptibility are compared, up to order ${\ensuremath{\beta}}^{6}$ for the ferromagnetic and antiferromagnetic phases, with the respective numerical results of a periodic chain with ten sites (N=10)(N=10)(N=10) and numerical results in the literature [Bl\"ote, Physica B 79, 427 (1975)]. A very good agreement is obtained for both phases in the high-temperature regime.
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We apply the cummulant method to obtain the high-temperature expansion of the Helmholtz free energy of the tetrahedral spin-1/2 and spin-2 XXZ models. The tetrahedral model is written as a composite spin-1 XXZ model, and some of its thermodynamic functions are compared to those of the ordinary spin-1 XXZ model. The composite spin-1 model is then mapped onto a fermion model, and it is shown that the contribution of the string Hamiltonian to thermodynamic functions at high temperatures cannot be neglected. The high-temperature expansion of the Helmholtz free energy of the anisotropic spin-222 XXZ chain is obtained up to order ${\ensuremath{\beta}}^{6}.$ Our results fit well the numerical quantum Monte Carlo data calculated by Yamamoto [Phys. Rev. B 53, 3364 (1996)] for the isotropic antiferromagnetic Heisenberg chain. We complement his high-temperature expansions for thermodynamic functions with terms of higher order in $\ensuremath{\beta}.$