Thermodynamics of Vortices in the Plane

Details Thermodynamics of Vortices in the Plane Thermodynamics of Vortices in the Plane

1993-07-27

arxiv.org/abs/hep-th/9307165v1

J.Math.Phys. 35 (1994) 1171-1184

Physics

High Energy Physics

High Energy Physics - Theory

17 pages, DAMTP 93-30

Scientific paper

10.1063/1.530633

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.

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