Quantum Statistical Mechanics of Soliton Bearing Systems. I: The Use of the Coherent State Representation

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Abstract
A quantum statistical mechanics of topological soliton bearing systems is developed by the use of the coherent state representation. This theory contains two parameters: R = (soliton width)/ (lattice spacing) and the measure of the quantum fluctuation Γ. The Bloch equation of the equilibrium density matrix in the coherent state representation is solved for the sine-Gordon system in the lowest order of Γ for the displacive limit R≫1. The transfer integral method can be applied to calculate the free energy similarly to the case of classical statistical mechanics. We can study the effect of finite temperatures successfully. In the present treatment the harmonic phonon is approximated by the Einstein phonon with frequency ω which is determined to be the Debye frequency essentially. The quantum effect on the soliton and the anharmonicity of phonon is described by a single factor g=(1-e-β \hbarω)/ β \hbarω. That is, the free energy is obtained by the replacement Es →√gEs in the classical expression of the nonlinear contribution to the free energy, where Es denotes the classical soliton creation energy. We examine the specific heat and a correlation length, finding considerable decreases for both quantities compared to the classical results. Such decrease of the specific heat is favourable for explaining experimental data.

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