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Brazilian Journal of Physics

Thermodynamics of the limiting cases of the XXZ model without Bethe ansatz

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

Published Dec 1, 2001• 4 citations
Publisher

Abstract

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.

DOI
10.1590/s0103-97332001000400008

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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 sub-chains. To exemplify its application we consider the Heisenberg XXZ model in two limit cases: the fermionized version Ising model (t = 0) and the free fermion case ( ∆ = 0). In both cases we recover the known expressions of the corresponding thermodynamical potential.
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.
Abstract We present a new algorithm to evaluate the grand potential at high and finite temperatures using many-body perturbation theory. This algorithm enables us to calculate the contribution of any Hugenholtz or Feynman vacuum diagrams and formulate the results as a sum of divided differences. Additionally, the proposed method is applicable to any interaction in any dimension, allowing us to calculate thermodynamic quantities efficiently at any given temperature, particularly at high temperatures.Furthermore, we apply this algorithm to the Heisenberg spin-1/2 XXZ chain. We obtain all coefficients of the high-temperature expansion of the free energy and susceptibility per site of this model up to the sixth order.
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.
We report on a new approach to the calculation of thermodynamic functions for crossing-invariant models solvable by Bethe Ansatz. In the case of the XXZ Heisenberg chain we derive, for arbitrary values of the anysotropy, a single non-linear integral equation from which the free energy can be exactly calculated.These equations are shown to be equivalent to an infinite set of algebraic equations of Bethe type which provide alternatively the thermodinamics. The high-temperature expansion follows in a sistematic and relatively simple way from our non-linear integral equations. For low temperatures we obtain the correct central charge and predict the analytic structure of the full expansion around T=0. Furthermore, we derive a single non-linear integral equation describing the finite-size ground-state energy of the Sine-Gordon quantum field theory.
We present a new approach to the calculation of thermodynamic functions for crossing-invariant models solvable by Bethe Ansatz. In the case of the XXZ Heisemberg chain we derive, for arbitrary values of the anysotropy, a {\bf single} non--linear integral equation from which the free energy can be exactly calculated. The high--temperature expansion follows in a sistematic and relatively simple way. For low temperatures we obtain the correct central charge and predict the analytic structure of the full expansion around T=0T=0T=0. Furthermore, we derive a single non-linear integral equation describing the finite--size ground--state energy of the Sine--Gordon quantum field theory. PACS: 05.30, 03.70. 75.10.5