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Physical Review A

Wigner-like expansion for the quantum-statistical mechanics of solids: Application to the sine-Gordon chain

Massimo Moraldi, A. Rettori, M. G. Pini

Published Mar 1, 1985• 13 citations
Publisher

Abstract

A semiclassical expansion for the quantum partition function of solids is presented. With respect to the usual Wigner method, the temperature range where the expansion is significant is broadened owing to the correct quantum-statistical treatment of the harmonic modes, while a Wigner-like expansion is retained only for the anharmonic part. As an application, the lowest quantum correction to the specific heat of a one-dimensional sine-Gordon model is calculated analytically.

DOI
10.1103/physreva.31.1971

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A semiclassical expansion for the quantum partition function of solids is presented. With respect to the usual Wigner method, the temperature range where the expansion is significant is broadened owing to the correct quantum-statistical treatment of the harmonic modes, while a Wigner-like expansion is retained only for the anharmonic part. As an application, the lowest quantum correction to the specific heat of a one-dimensional sine-Gordon model is calculated analytically.
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.
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 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.
The equilibrium quantum statistics of various anharmonic oscillator systems including relativistic systems is considered within the Wigner phase space formalism. For this purpose the Wigner series expansion for the partition function is generalized to include relativistic corrections. The new series for partition functions and all thermodynamic potentials yield quantum corrections in terms of powers of h(sup 2) and relativistic corrections given by Kelvin functions (modified Hankel functions) K(sub nu)(mc(sup 2)/kT). As applications, the symmetric Toda oscillator, isotonic and singular anharmonic oscillators, and hindered rotators, i.e. oscillators with cosine potential, are addressed.
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 for the quantum partition function of a chain of atoms with nearest-neighbor interaction is calculated by a variational method. This approach turns out to be very rapid and powerful; it is able to describe the quantum properties of the system in the ``solid'' phase. Application with the Lennard-Jones potential is made and explicit results for internal energy and specific heat are shown. Our technique based on the effective potential offers an alternative to the quantum Monte Carlo approach.