Free energies for the discrete chain in a periodic potential and the dual Coulomb gas

Publisher Website
Google Scholar
Abstract
The partition function for a chain of particles connected by springs in a periodic potential is considered. This problem is dual to a one-dimensional Coulomb gas on a lattice. The free energies of both problems can be calculated from the eigenvalue of a transfer matrix. High-temperature expansions are obtained for free energies of both problems. The free energy of a system of alternating charges on a lattice is calculated exactly. Continuum results for the Coulomb gas and the sine-Gordon model are easily regained from the transfer-matrix approach. For the Villain potential the partition function can be written directly in terms of the kinks in the chain. The kinks are on sites of a lattice and interact through an exponential repulsion. The ground-state periodicity of this system exhibits a complete devil's staircase as a function of mismatch. For a similar potential the free energy can be calculated at all temperatures as the eigenvalue of a differential equation. A ladder of Josephson junctions is proposed as a new physical realization for this problem.