We investigate, via Monte Carlo simulations, the phase structure of a system of closed, nonintersecting but otherwise noninteracting, loops in 3 Euclidean dimensions. The loops correspond to closed trajectories of massive particles and we find a phase transition as a function of their mass. We identify the order parameter as the average length of the loops at equilibrium. This order parameter exhibits a sharp increase as the mass is decreased through a critical value, the behavior seems to be a crossover transition. We believe that the model represents an effective description of the broken-symmetry sector of the $2+1$ dimensional Abelian Higgs model, in the extreme strong coupling limit. The massive gauge bosons and the neutral scalars are decoupled, and the relevant low-lying excitations correspond to vortices and antivortices. The functional integral can be approximated by a sum over simple, closed vortex loop configurations. We present a novel fashion to generate nonintersecting closed loops, starting from a tetrahedral tessellation of three space. The two phases that we find admit the following interpretation: the usual Higgs phase and a novel phase which is heralded by the appearance of effectively infinitely long loops. We compute the expectation value of the Wilson loop operator and that of the Polyakov loop operator. The Wilson loop exhibits perimeter law behavior in both phases implying that the transition corresponds neither to the restoration of symmetry nor to confinement. The effective interaction between external charges is screened in both phases, however there is a dramatic increase in the polarization cloud in the novel phase as shown by the energy shift introduced by the Wilson loop.

• Received 17 October 2007

DOI:https://doi.org/10.1103/PhysRevD.81.114505