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Physical review. B, Condensed matter

Variational approach to the thermodynamics of a quantum sine-Gordon field with out-of-plane fluctuations

R. Giachetti, V. Tognetti

Published Oct 1, 1987• 17 citations
Publisher

Abstract

The quantum corrections to the partition function of nonlinear systems with a nonlocal kinetic energy are calculated by extending a variational approach based on the path integral and developed in a previous paper, which allows one to take into account in the quantum scheme of the quadratic part of the Hamiltonian. This extension can be useful to study the role of the out-of-plane fluctuations which cause deviations from the sine-Gordon model for some magnetic chains.

DOI
10.1103/physrevb.36.5512

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The quantum corrections to the partition function of nonlinear systems with a nonlocal kinetic energy are calculated by extending a variational approach based on the path integral and developed in a previous paper, which allows one to take into account in the quantum scheme of the quadratic part of the Hamiltonian. This extension can be useful to study the role of the out-of-plane fluctuations which cause deviations from the sine-Gordon model for some magnetic chains.
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.
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.
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 classical effective Hamiltonian for the quantum easy-plane ferromagnetic chain with an in-plane applied field is introduced by means of a path-integral variational method. It then becomes possible to include quantum effects of the out-of-plane fluctuations in an easily evaluable phase-space integral which approximates the quantum partition function. A comparison with experimental data for the excess specific heat of CsNiF3{\mathrm{CsNiF}}_{3}CsNiF3​ shows much improvement in the description of this system. In addition, new light is shed on the old problem of defining the classical counterpart of a spin Hamiltonian.