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Physica Scripta

Effective potential for the quantum thermodynamics of integrable and non integrable one-dimensional systems

R. Giachetti, V. Tognetti, Ruggero Vaia

Published Oct 1, 1989• 2 citations
Publisher

Abstract

A variational approach based on the path-integral formulation of the statistical mechanics is applied to calculate the partition function and the related thermodynamic quantities of one-dimensional kink bearing fields. This is done by determining an effective potential which includes in a complete quantum way the linear modes of the field, while treating variationally the nonlinear excitations. The treatment is applied both to integrable systems, like Sine-Gordon and to non integrable ones, like φ 4 and Double Sine-Gordon. The temperature renormalization can be separately studied both for the vacuum and the one-soliton sector in order to find the low temperature properties in self consistent one-loop approximation. Moreover a low-coupling expansion is given and its range of applicability is found to be much wider than the Wigner expansion. Comparisons with quantum Monte-Carlo results and exact results obtained by Bethe Ansatz are presented.

DOI
10.1088/0031-8949/40/4/002

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A variational approach based on the path-integral formulation of the statistical mechanics is applied to calculate the partition function and the related thermodynamic quantities of one-dimensional kink bearing fields. This is done by determining an effective potential which includes in a complete quantum way the linear modes of the field, while treating variationally the nonlinear excitations. The treatment is applied both to integrable systems, like Sine-Gordon and to non integrable ones, like φ 4 and Double Sine-Gordon. The temperature renormalization can be separately studied both for the vacuum and the one-soliton sector in order to find the low temperature properties in self consistent one-loop approximation. Moreover a low-coupling expansion is given and its range of applicability is found to be much wider than the Wigner expansion. Comparisons with quantum Monte-Carlo results and exact results obtained by Bethe Ansatz are presented.
A variational approach based on the path-integral formulation of the statistical mechanics is applied for calculating the partition function of the sine-Gordon field. This is done by determining an effective potential which includes, in a complete quantum way, the linear modes of the field, while treating variationally the nonlinear excitations. Using this effective potential the temperature renormalization is separately studied both for the vacuum and the one-soliton sector, recovering the total one-loop self-consistent (Hartree-Fock) renormalized approximation. A high-temperature expansion is calculated whose range of applicability is found to be much wider with respect to previous expansions of the same kind. Finally, a comparison with a Monte Carlo simulation is presented, obtaining a very good agreement with the numerical data.
The quantum thermodynamics of the nonintegrable ${\ensuremath{\varphi}}^{4}$ one-dimensional chain is studied by means of a classical effective potential, which includes in a fully quantum way the linear modes of the field. In contrast to the case of the sine-Gordon chain, integrable in the continuum limit, exact results are not available, so that approximate quantum calculations appear to be useful. This effective potential is determined by a variational approach developed in previous papers, which is based on the path-integral formulation of statistical mechanics. The temperature renormalization is studied, in the limit of low temperature, by means of a self-consistent saddle-point method, both for the vacuum and the one-kink sectors, and the results of the semiclassical approximation are recovered. Important results are obtained by a new low-coupling expansion for the effective potential. Its range of validity in temperature is much wider than the range of previous high-temperature expansions. The results for the nonlinear contributions to internal energy and specific heat, obtained by means of original transfer-matrix computations, are finally presented and discussed.
The path-integral method is used for determination of the quantum corrections to the free energy of nonlinear systems. All quantum effects of the harmonic part of the potential are considered and a variational principle is used to account for the quantum corrections due to the anharmonic part. Correct renormalized frequencies are obtained at any temperature and an effective potential to be inserted in the configurational integral is found. A new general expression for the partition function at any temperature in the low-coupling limit is obtained.
The path-integral approach is used for determining quantum corrections to the free energy of nonlinear systems. An effective potential, to be inserted in the configurational integral, is constructed by a new variational scheme which gives the quantum modification of the potential due to the anharmonic part together with the frequency renormalization. The single anharmonic oscillator is studied first. Then the extension to fields is presented and a new general expression for the partition function is given. The method is applied to the sine-Gordon chain and explicit calculations of the specific heat are shown.