Thermodynamics of Vortices in the Plane

TL;DR

This paper studies the thermodynamics of vortices in the abelian Higgs model on a plane with periodic boundary conditions, deriving the equation of state and analyzing the moduli space volume without finding phase transitions.

Contribution

It provides a detailed geometric analysis of the vortex moduli space and derives the vortex equation of state in the thermodynamic limit.

Findings

The vortex moduli space volume factors into base and fiber volumes.

The equation of state is $P(A-4 ewline ext{pi} N)=NT$.

No phase transition occurs in the vortex system.

Abstract

The thermodynamics of vortices in the critically coupled abelian Higgs model, defined on the plane, are investigated by placing $N$ vortices in a region of the plane with periodic boundary conditions: a torus. It is noted that the moduli space for $N$ vortices, which is the same as that of $N$ indistinguishable points on a torus, fibrates into a $CP_{N-1}$ bundle over the Jacobi manifold of the torus. The volume of the moduli space is a product of the area of the base of this bundle and the volume of the fibre. These two values are determined by considering two 2-surfaces in the bundle corresponding to a rigid motion of a vortex configuration, and a motion around a fixed centre of mass. The partition function for the vortices is proportional to the volume of the moduli space, and the equation of state for the vortices is $P(A-4\pi N)=NT$ in the thermodynamic limit, where $P$ is the pressure, $A$ the area of the region of the plane occupied by the vortices, and $T$ the temperature. There is no phase transition.