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Physical review. B, Condensed matter

Exact Bethe-ansatz thermodynamics for the sine-Gordon model in the classical limit: Effect of long strings

JUSSI TIMONEN, R. Bullough, D. J. Pilling

Published Nov 1, 1986• 31 citations

Abstract

We solve the finite-temperature Bethe-ansatz (BA) integral equations with coupling constant $\ensuremath{\mu}=\ensuremath{\pi}(1\ensuremath{-}{n}^{\ensuremath{-}1})+0$, in the semiclassical limit $n=\mathrm{integer}\ensuremath{\rightarrow}\ensuremath{\infty}$. After eliminating long strings and then an exact bosonization, transforming $n\ensuremath{-}2$ fermion strings into a single boson, the BA integral equations are reduced to two, coupling bosons and two fermions (kinks and antikinks). In the classical limit these iterate exactly to the sine-Gordon free energy found by the transfer integral method and by other methods recently reported. We suggest that our result can be extended to general values of $\ensuremath{\mu}$.

DOI: 10.1103/physrevb.34.6525

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We solve the finite-temperature Bethe-ansatz (BA) integral equations with coupling constant

\ensuremath{\mu}=\ensuremath{\pi}(1\ensuremath{-}{n}^{\ensuremath{-}1})+0

, in the semiclassical limit

n=\mathrm{integer}\ensuremath{\rightarrow}\ensuremath{\infty}

. After eliminating long strings and then an exact bosonization, transforming

n\ensuremath{-}2

fermion strings into a single boson, the BA integral equations are reduced to two, coupling bosons and two fermions (kinks and antikinks). In the classical limit these iterate exactly to the sine-Gordon free energy found by the transfer integral method and by other methods recently reported. We suggest that our result can be extended to general values of

\ensuremath{\mu}

.

We use the quantum Bethe ansatz method to compute the free energy of the sine-Gordon model in the classical limit. In this limit the number of breathers, and hence the number of coupled integral equations to be solved, diverges. By linearizing the breather mass spectrum and phase shifts, extending a method of Maki, we can reduce the breather ladder to anharmonic phonons. The divergent set of integral equations is reduced to only two, for interacting phonons and solitons. We solve these equations iteratively to give the free energy in a double series in the temperature t and the soliton density ${e}^{\mathrm{\ensuremath{-}}1/t}$ which agrees to high order with classical transfer integral results.

We use the quantum Bethe-Ansatz method to compute the free energy of the classical sine-Gordon system. Previous attempts to do this have failed because the number of coupled integral equations to be solved to find the free-energy diverges in the classical limit. We present a transformation, extending a method of Maki, which reduces this divergent set to only two coupled equations, for the densities of solitons and anharmonic phonons. These equations can be solved iteratively in the temperature

t

and the soliton density ${e}^{\frac{\ensuremath{-}1}{t}}$ to give a double series for the free energy. This series coincides to high order with classical transfer-matrix results.

The Bethe-ansatz (BA) equation for the classical sine-Gordon (SG) thermodynamics is shown to reduce to the one for the classical Toda lattice within two approximations; the phonon-phonon interactions are neglected (the harmonic phonon) and solitons are treated nonrelativistically. Within these approximations the solitons in the SG model correspond to the particles in the Toda lattice. Making use of an analytical solution of the BA equation for the classical Toda lattice due to Opper, we obtain an analytical solution of the approximate SG equation. We propose also a similar approximate BA equation for the ${\ensuremath{\varphi}}^{4}$ model, although the ${\ensuremath{\varphi}}^{4}$ model does not have the BA equation for the classical thermodynamics.

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

n

, 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

\mathrm{XYZ}

spin chain is presented, and a non-Lorentz-invariant feature of that limit is noted.