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Physical review. B, Condensed matter

Analytical results for the high-temperature expansion of the one-dimensional s=1XXZ model

Onofre Rojas, E. V. Corrêa Silva, W. A. Moura-Melo, S. M. de Souza, M. T. Thomaz

Published Mar 31, 2003• 4 citations
Publisher

Abstract

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-$1$ $\mathrm{XXZ}$ 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 $(N=10)$ 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.

DOI
10.1103/physrevb.67.115128

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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.
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.
The Heisenberg XXZ model is a chain model with nearest-neighbor interactions. Its thermodynamics is exactly obtained via Bethe ansatz. Recently, we developed a method to derive the high-temperature expansion of the grand potential per site of translationally invariant chain models, with periodic boundary conditions. Here we apply this approach to the XXZ model with periodic boundary conditions for the Ising limit case (t = 0) and the free fermion case (delta = 0 and h = 0), obtaining results in agreement with the literature. In this new way of obtaining the coefficients of the high-temperature expansion of the grand potential, the coefficients are derived from an auxiliary function written only in terms of open connected sub-chains.
We investigate the grand potential of the one-dimensional Hubbard model in the high-temperature limit, calculating the coefficients of the high-temperature expansion $(\ensuremath{\beta}$-expansion) of this function up to order ${\ensuremath{\beta}}^{4}$ by an alternative method. The results derived are analytical and do not involve any perturbation expansion in the hopping constant, being valid for arbitrary density of electrons in the one-dimensional model. In the half-filled case, we compare our analytical results for the specific heat and the magnetic susceptibility, in the high-temperature limit, with the ones obtained by Beni et al. [Phys. Rev. B 8, 3329 (1973)] and Takahashi's integral equations, showing that the latter result does not take into account the complete energy spectrum of the one-dimensional Hubbard model. The exact integral solution by J\"uttner et al. [Nucl. Phys. B 522, 471 (1998)] is applied to the determination of the range of validity of our expansion in $\ensuremath{\beta}$ in the half-filled case, for several different values of U.
We consider an arbitrary translationally invariant chain model with nearest neighbors interaction and satisfying periodic boundary condition. The approach developed here allows a thermodynamic description of the chain model directly in terms of grand potential per site. This thermodynamic function is derived from an auxiliary function constructed only from open connected subchains. In order to exemplify its application and how this approach works we consider the Heisenberg XXZ model. We obtain the coefficients of the high temperature expansion of the free energy per site of the model up to third order.