Phase transitions in a vortex gas

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 ≫ λ the equation of state is of the Van der Waals form ( P + bN²/ A²) ( A - aN) = NT with a = 4 π and b = -4.89 πλ where λ is a measure of the distance from critical coupling. It is found that there is no phase transition in a low-density type-I1 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.