Phase transitions at strong coupling in the 2+1-d abelian Higgs model

We study, using numerical Monte-Carlo simulations, an effective description of the 2+1 dimensional Abelian Higgs model which is valid at strong coupling, in the broken symmetry sector. In this limit, the massive gauge boson and the massive neutral Higgs decouple leaving only the massive vortices. The vortices have no long range interactions. We find a phase transition as the mass of the vortices is made lighter and lighter. At the transition, the contributions to the functional integral come from a so-called infinite vortex anti-vortex loop. Adding the Chern-Simons term simply counts the linking number between the vortices. We find that the Wilson loop exhibits perimeter law behaviour in both phases, although the polarization cloud increases by an order of magnitude at the transition. We also study the 't Hooft loop. We find the 't Hooft loop exhibits perimeter law behaviour in the presence of the Chern-Simons term but is trivial in its absence. Thus we have a theory with perimeter law for both the Wilson loop and the 't Hooft loop, but contains no massless particles.


Abelian Higgs model at strong coupling
The abelian Higgs model corresponds to the Lagrangian: At strong coupling, λ, e → ∞, keeping λ/e 2 fixed, the neutral scalar mass √ 2λη and the gauge boson mass eη diverge but the vortex mass μ = η 2 f (2λ/e 2 ) can be held finite and variable, see [1] for details.

Evidence for a phase change
We calculate the Euclidian vacuum persistence amplitude using the functional integral and lattice Monte Carlo simulations. The field configurations that contribute to this amplitude correspond to closed vortex loops, which describe the virtual process of the creation of a vortex anti-vortex pair and its subsequent annihilation. The Euclidian action for such a configuration is, to first approximation, simply the vortex mass multiplied by the length of the vortex loop [2]. Thus we generate configurations of closed loops on a 3 dimensional lattice, and weigh each such configuration using the Boltzmann factor corresponding to the total length of the loops multiplied by μ. For large μ only small minimal loops enter the calculation, however as μ is lowered, larger and larger loops become important. After a specific value, μ = .152, there is a sudden transition where about half of the loops reorganize and form a so-called infinite loops. for smaller values of μ, the total length of the vortex loops increases, however this occurs simply by appending to the length of the infinite loop, the number of finite loops nor their total length do not appreciably change.
An infinite loop corresponds to a loop that has a length substantially larger than what a loop should have if it were of a linear dimension equal to the size of the lattice. The loops are closed non-self intersecting random walks. On a cubic lattice their linear dimension should be approximately N (3/5) [3]. Our lattice is actually a much more intricate lattice, where it is not known what the size of an infinite loop should be. We consider any loop longer than 10000 steps to be infinite.
Clear evidence for a change of phase is seen in Figure (1) of the average length of the loops versus the logarithm of the mass of the vortices, ln μ. We see that as μ is diminished, the average length of the loops increases dramatically about six-fold.
We then evaluate this for various different spatial sizes, we find for example: which show a perfect

Chern-Simons term
It is possible to add the Chern-Simons term to the action: The Chern-simons term converts charged particles or magnetic flux tubes into anyons. Hence the vortices behave as fractional statistics particles. With this normalization, the term is just equal to 2κ × L T where L T is the total linking number of all the vortex loops with each other. The important distinction is that in Euclidean space this term, being t-odd remains imaginary [5] in Euclidean space.. Hence it cannot be used to define the Boltzmann factor in the Monte Carlo simulation. However the real part of the action provides a perfectly good measure on the space In principle, the set of Green's functions obtained from this measure will satisfy the Wightman axioms [6], and can be used to reconstruct [7] the full Chern-Simons added quantum field theory.

Wilson loop and 't Hooft loop in the Chern-Simons theory
The calculation of these two order parameters follows in a straightforward fashion in the Chern-Simons theory, for details see [8].

Conclusions
We have computed using Monte Carlo simulations in an effective description of the 2+1 dimensional Abelian Higgs model, properties of the vacuum. Although we find a phase change, the expectation value of the Wilson loop and the 't Hooft loop does not change. The 't Hooft loop is trivial without the Chern-simons term, the Wilson loop displays perimeter law behaviour in both phases. The two order parameters continue to display the perimeter law in both phases, even in the presence of the Chern-Simons term. This is a counter example to the understanding that such behaviour necessitates the existence of massless particles, [10]. Here we have no massless particles, however the Chern-Simons term exercises a subtle long range, statistics changing interaction, which is able to permit the perimeter law behaviour for both order parameters.