Superconducting networks and superfluid films in two dimensions are often described by a theoretical model in which the unique microscopic variables are phases. Among these models the XY model with Villain’s interaction potential can be mapped exactly onto a lattice Coulomb gas. This is well known, but several questions still have no clear answers: First, what is the meaning of the charge of the Coulomb gas in terms of the original variables of the XY model? Second, how can the helicity modulus be expressed exactly in the Coulomb gas representation on a finite torus? In this paper we answer these questions. The mapping onto a lattice Coulomb gas is done in a way that differs from the usual one. This mapping is applied to a phase model whose partition function has an identical mathematical structure as the one of the XY model with Villain’s interaction. For this phase model, contrary to the XY model, the charges of the Coulomb gas describe indeed exactly the topological charges as we can define them in terms of the phase variables. However, this Coulomb gas contains an additional polarization energy and two additional fictitious variables accounting for the specific topological character of the torus. The helicity modulus is exactly the inverse of a dielectric constant which can be defined as the linear response to an external uniform electric field, even on a torus. The meaning of the Coulomb-gas representation is also discussed in terms of the original variables of the XY model.

• Received 27 January 1994

DOI:https://doi.org/10.1103/PhysRevB.50.4015