Percolation in a self-avoiding vortex gas model of the lambda transition in three dimensions

A Monte Carlo simulation of the equilibria of a three-dimensional gas of vortex loops is conducted. Lines of vorticity are expressed as closed self-avoiding walks on a lattice. Vortices interact at low temperatures through a Biot-Savart potential arising through the Villian approximation of the XY action. The vortex model at infinite temperature is also related to Symanzik's random-loop expansion of the critical XY model. The low-temperature state of the model simulates thermally excited vorticity in superfluid helium. The high temperature state of the vortex gas has application to classical turbulence. A grand-canonical ensemble is simulated, in which chemical potential is a parameter related to the self-energy per unit length of a vortex line. Two phases are identified in the temperature-chemical potential plane; the phases are separated by a transition line analogous to the lambda transition of superfluid helium. The transition line is also a percolation threshold; all vortex loops are of finite size in the cold phase, while an infinite vortex loop exists in the hot phase. The percolation transition driven by chemical potential persists to infinite temperature. At the infinite temperature ''polymeric'' state the Biot-Savart interaction vanishes and the behavior of the system is determined only by the density of vortices and geometric constraints.