The sine-Gordon chain: Equilibrium statistical mechanics

Abstract

The equilibrium statistical mechanics of the sine-Gordon chain is studied using transfer-integral techniques. Implementation of formally exact expressions for the free energy, equilibrium averages, etc., rests upon solution of the transfer-integral (TI) problem. Away from the continuum limit the solution to the TI problem contains features that must be examined with some care. Several approximate schemes for solving the TI problem are described and subjected to numerical verification. The appropriate thermodynamic variables for describing the sine-Gordon chain are found to be temperature and phase; the mechanical variable conjugate to phase is torque. Examination of the Helmholtz free energy, torque, equilibrium averages, etc., shows that as $T\to 0^{+}$ the chain with nonzero phase is described by a gas of noninteracting solitons. At $T=0\mathrm{}°$K the chain with nonzero phase corresponds to a kink lattice. At $T=0^{+}$ this lattice "melts" although the phase evolution continues to occur locally on the chain as solitons.