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Journal of the Physical Society of Japan

Low-Temperature Expansions and the Euler-Maclaurin Formula

Hideyuki Mizuta, Miki Wadati

Published Apr 1, 1997• 2 citations
Publisher

Abstract

Thermodynamic quantities of solvable models such as one–dimensional Heisenberg model can be obtained through the finite temperature Baxter formula. For the low–temperature expansions, the estimate of the finite size corrections by use of the Euler–Maclaurin formula is essential. There has been a question on the convergence of such calculations. In this paper, choosing the XX model as a simple example, we show explicitly that the Euler–Maclaurin formula reproduces the correct low–temperature expansion of the free energy.

DOI
10.1143/jpsj.66.988

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Thermodynamic quantities of solvable models such as one–dimensional Heisenberg model can be obtained through the finite temperature Baxter formula. For the low–temperature expansions, the estimate of the finite size corrections by use of the Euler–Maclaurin formula is essential. There has been a question on the convergence of such calculations. In this paper, choosing the XX model as a simple example, we show explicitly that the Euler–Maclaurin formula reproduces the correct low–temperature expansion of the free energy.
Thermodynamic quantities of solvable models such as one–dimensional Heisenberg model can be obtained through the Finite temperature Baxter formula. For the low–temperature expansions, the estimate of the finite size corrections by use of the Euler–Maclaurin formula is essential. There has been a question on the convergence of such calculations. In this paper, we confirm that the Euler–Maclaurin formula gives the correct low–temperature expansion of the free energy for the XXZ Heisenberg model.
Recent developments in the theory of exactly solvable models are reviewed. Particular attention is paid to the finite size corrections to the Bethe ansatz equations. Baxter’s formula which relates a 2-dimensional statistical model with a 1-dimensional spin model is extended into the finite temperature case. A combination of this extension and the theory of finite size corrections gives a systematic method to evaluate low temperature expansions of physical quantities. Applications of the method to 1-dimensional quantum spin models are discussed. Throughout this paper, the usefulness of the soliton theory should be observed.
Abstract The Bethe ansatz genuinely considers a finite system. The extraction of finite-size results from the Bethe ansatz equations is of genuine interest, especially against the background of the results of finite-size scaling and conformal symmetry in finite geometries. The mathematical techniques introduced in chapter 19 permit a systematic treatment in this chapter of finite-size corrections as corrections to the thermodynamic limit of the system. The application of the Euler-Maclaurin formula transforming finite sums into integrals and finite-size corrections transforms the Bethe ansatz equations into Wiener–Hopf integral equations with inhomogeneities representing the finite-size corrections solvable using the Wiener–Hopf technique. The results can be compared to results for finite systems obtained from other approaches that are independent of the Bethe ansatz method. It briefly discusses higher-order corrections and offers a general assessment of the finite-size method.
High-temperature expansions performed at a fixed-order parameter provide a simple and systematic way to derive and correct mean-field theories for statistical mechanical models. For models like spin glasses which have general couplings between spins, the authors show that these expansions generate the Thouless-Anderson-Palmer equations at low order. They explicitly calculate the corrections to TAP theory for these models. For ferromagnetic models, they show that their expansions can easily be converted into 1/d expansions around mean-field theory, where d is the number of spatial dimensions. Only a small finite number of graphs need to be calculated to generate each order in 1/d for thermodynamic quantities like free energy or magnetization. Unlike previous 1/d expansions, the expansions are valid in the low-temperature phases of the models considered. They consider alternative ways to expand around mean-field theory besides 1/d expansions. In contrast to the 1/d expansion for the critical temperature, which is presumably asymptotic, these schemes can be used to devise convergent expansions for the critical temperature. They also appear to give convergent series for thermodynamic quantities and critical exponents. They test the schemes using the spherical model, where their properties can be studied using exact expressions.