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Physical Review A

Continuum limit of sine-Gordon and φ4 chains by use of an effective potential

R. Giachetti, V. Tognetti, Ruggero Vaia

Published Aug 1, 1988• 21 citations
Publisher

Abstract

The effective potential previously introduced by the variational method is used to calculate thermodynamic quantities of quantum one-dimensional nonlinear fields in the weak-coupling approximation. The correct continuum limit is recovered both for integrable (sine-Gordon) and noninte- grable (${\ensuremath{\varphi}}^{4}$) models. Good agreement is found with exact Bethe-ansatz results for the specific heat.

DOI
10.1103/physreva.38.1638

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The effective potential previously introduced by the variational method is used to calculate thermodynamic quantities of quantum one-dimensional nonlinear fields in the weak-coupling approximation. The correct continuum limit is recovered both for integrable (sine-Gordon) and noninte- grable (\ensuremathφ4{\ensuremath{\varphi}}^{4}\ensuremathφ4) models. Good agreement is found with exact Bethe-ansatz results for the specific heat.
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.
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.
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.
The Bethe-ansatz equations for the thermodynamic properties of the quantum sine-Gordon systems are derived in the zero-charge-sector attractive case. For rational values of the coupling parameter $\frac{\ensuremath{\mu}}{\ensuremath{\pi}}$ these reduce to a finite set, solved here numerically for $\ensuremath{\mu}=[\frac{(n\ensuremath{-}1)}{n}]\ensuremath{\pi}$, for several values of nnn, to give the specific heat as a function of temperature. The "soliton" contribution peaks at $\ensuremath{\simeq}0.4$ soliton masses for $\ensuremath{\mu}=\frac{4}{5}\ensuremath{\pi}$, shifting downward for higher $\ensuremath{\mu}$. A detailed analysis of the sine-Gordon limit of the XYZ\mathrm{XYZ}XYZ spin chain is presented, and a non-Lorentz-invariant feature of that limit is noted.