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Physical review. B, Condensed matter

Bethe-ansatz quantum sine-Gordon thermodynamics. The specific heat

Michael Fowler, X. Zotos

Published May 1, 1982• 70 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.25.5806

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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 \ensuremathμ\ensuremathπ\frac{\ensuremath{\mu}}{\ensuremath{\pi}}\ensuremathπ\ensuremathμ​ these reduce to a finite set, solved here numerically for \ensuremathμ=[(n\ensuremath−1)n]\ensuremathπ\ensuremath{\mu}=[\frac{(n\ensuremath{-}1)}{n}]\ensuremath{\pi}\ensuremathμ=[n(n\ensuremath−1)​]\ensuremathπ, for several values of nnn, to give the specific heat as a function of temperature. The "soliton" contribution peaks at \ensuremath≃0.4\ensuremath{\simeq}0.4\ensuremath≃0.4 soliton masses for \ensuremathμ=45\ensuremathπ\ensuremath{\mu}=\frac{4}{5}\ensuremath{\pi}\ensuremathμ=54​\ensuremathπ, shifting downward for higher \ensuremathμ\ensuremath{\mu}\ensuremathμ. 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 show that the thermodynamic properties of the quantum sine-Gordon system in the zero-charge sector can be found from the appropriate limit of the Takahashi-Suzuki Bethe ansatz analysis of the XYZ\mathrm{XYZ}XYZ spin chain. We prove that the rather complex criteria for allowed states in thermodynamic sums in that analysis correspond simply to including only states with normalizable wave functions. The behavior of the XYZ\mathrm{XYZ}XYZ phase-shift function in the sine-Gordon limit is discussed.
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.
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.