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Journal of Physics A Mathematical and General

Is the finite-temperature sine-Gordon soliton mass discontinuous?

Michael Fowler, M D Johnson

Published Nov 21, 1988• 1 citation
Publisher

Abstract

The integrability of a quantum sine-Gordon model allows finite densities of zero binding energy 'bound states' at finite temperatures for certain values of the coupling constant. At these same coupling strengths, the usual Yang and Yang formulation (1969) of Bethe ansatz thermodynamics appears to predict discontinuities in some elementary excitation masses, for example solitons. This is examined in two limits-free fermions and free bosons. The apparent discontinuity turns out to be an artefact of the formalism, common to all cases where zero binding energy bound states exist.

DOI
10.1088/0305-4470/21/22/017

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The integrability of a quantum sine-Gordon model allows finite densities of zero binding energy 'bound states' at finite temperatures for certain values of the coupling constant. At these same coupling strengths, the usual Yang and Yang formulation (1969) of Bethe ansatz thermodynamics appears to predict discontinuities in some elementary excitation masses, for example solitons. This is examined in two limits-free fermions and free bosons. The apparent discontinuity turns out to be an artefact of the formalism, common to all cases where zero binding energy bound states exist.
For the quantum sine-Gordon--massive Thirring model at finite temperature, the relationship between the Bethe ansatz ``elementary excitations'' and physical excitations of the system is examined. The thermodynamic equations are written in terms of a set of dressed particles in the general case. We find that the recently predicted soliton mass discontinuity (on varying coupling strength at finite temperature) is not a real effect--the function exhibiting the discontinuity does not correspond to the mass of the soliton.
The thermal renormalization of the Bethe-ansatz excitation spectrum for the sine-Gordon system is presented. For the soliton mass, the results agree closely with those of Maki and Takayama over a significant temperature range, but the sudden decrease in mass they predict near a particular temperature is not found. For the Bethe-ansatz excitation corresponding to the phonon at zero temperature, the rest mass changes rapidly as the temperature increases, even near the classical limit. This suggests that at finite temperature this excitation (the one-string) no longer corresponds to the physical phonon.
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.
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 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 ttt 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.