Phase Transitions in a Vortex Gas

TL;DR

This paper models a vortex gas in the Abelian Higgs model using a particle dynamics approximation, deriving an equation of state and analyzing phase transitions, especially noting differences between type-I and type-II regimes.

Contribution

It applies a particle dynamics approximation to derive the equation of state for a vortex gas, including virial expansion and phase transition analysis.

Findings

Low-density equation of state resembles Van der Waals form.

No phase transition in low-density type-II gas.

Phase transition occurs in type-I gas between condensed and gaseous states.

Abstract

It has been shown recently that the motion of solitons at couplings around a critical coupling can be reduced to the dynamics of particles (the zeros of the Higgs field) on a curved manifold with potential. The curvature gives a velocity dependent force, and the magnitude of the potential is proportional to the distance from a critical coupling. In this paper we apply this approximation to determining the equation of state of a gas of vortices in the Abelian Higgs model. We derive a virial expansion using certain known integrals of the metric, and the second virial coefficient is calculated, determining the behaviour of the gas at low densities. A formula for determining higher order coefficients is given. At low densities and temperatures $T \gg \l$ the equation of state is of the Van der Waals form $(P+b\frac{N^{2}}{A^{2}})(A-aN) = NT$ with $a=4\pi$ and $b=-4.89\pi\l$ where $\l$ is a measure of the distance from critical coupling. It is found that there is no phase transition in a low density type-II gas, but there is a transition in the type-I case between a condensed and gaseous state. We conclude with a discussion of the relation of our results to vortex behaviour in superconductors.