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Journal of Physics A Mathematical and General

Quantum and classical statistical mechanics of the sinh-Gordon equation

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

Published Nov 11, 1986• 25 citations
Publisher

Abstract

The authors give two fundamental methods for quantum or classical free energies of integrable models. Periodic boundary conditions induce an integral equation for classically allowed momenta. Generalisations of the Berthe ansatz and a method of functional integration on the classical action in action-angle variables follow, giving identical (Bose-Fermi equivalent) results. For sinh-Gordon the Bose classical limits agree with the transfer integral method.

DOI
10.1088/0305-4470/19/16/002

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The authors give two fundamental methods for quantum or classical free energies of integrable models. Periodic boundary conditions induce an integral equation for classically allowed momenta. Generalisations of the Berthe ansatz and a method of functional integration on the classical action in action-angle variables follow, giving identical (Bose-Fermi equivalent) results. For sinh-Gordon the Bose classical limits agree with the transfer integral method.
We give two fundamental methods for evaluation of classical free energies of all the integrable models admitting soliton solutions; the sine-Gordon equation is one example. Periodic boundary conditions impose integral equations for allowed phonon and soliton momenta. From these, generalized Bethe-Ansatz and functional-integration methods using action-angle variables follow. Results for free energies coincide, and coincide with those that we find by transfer-integral methods. Extension to the quantum case, and quantum Bethe Ansatz, on the lines to be reported elsewhere for the sinh-Gordon equation, is indicated.
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 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 $\ensuremath{\mu}$.
Recent developments in statistical mechanics of quantum integrable systems are reviewed. Those studies are fundamental and have a renewed interest related to newly developing fields such as atomic Bose-Einstein condensations, photonic crystals and quantum computations. After a brief summary of the basic concepts and methods, the following three topics are discussed. First, by the thermal Bethe ansatz (TBA), a hard-core Bose gas is exactly solved. The model includes fully the effect of excluded volume and is identified to be a c=1 conformal field theory. Second, the cluster expansion method based on the periodic boundary condition for the Bethe wave function, which we call the Bethe ansatz cluster expansion (BACE) method, is developed for a δ-function gas and the XXX Heisenberg chain. This directly proves the TBA and reveals intrinsic properties of quantum integrable systems. Third, for a δ-function gas, the integral equations for the distribution functions of the quasi-momentum and the quasi-particle energy are solved in the form of power series. In the weak coupling case, the results reproduce those of Bogoliubov theory.
The calculation of the equilibrium free energy of integrable models like the sine-Gordon and attractive nonlinear Schrödinger models is discussed in the context of biological molecules like DNA : the thermalisation process (approach to equilibrium) is also discussed. The sine-Gordon model has a "repulsive" form which is the sinh-Gordon model. The approach to equilibrium of the sinh-Gordon model is described in all completeness in terms of a quantum mechanical master equation at finite temperatures. Although the dynamical evolution of the master equation as written is a solved problem, only the equilibrium solution is examined in this paper. The equilibrium free energy is calculated exactly as an integral equation for certain excitation energies at finite temperatures. Bose-fermi equivalent forms of this integral equation are given. The bose form yields a similar integral equation in classical limit. The iteration of this yields a low temperature asymptotic series for the classical free energy which checks against the result of the transfer integral method (TIM). Results for the zero temperature quantum eigenenergies are found. A further discussion of the dynamics of the approach to thermal equilbrium is made.
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.