Abstract

Vortex lines are a feature of many random or disordered three-dimensional systems (e.g. the 3D XY model). They show continuous percolation-like phase transitions, separating phases with only short vortex loops from phases with infinite vortex lines. This talk will begin with the problem of constructing a field theory for such transitions. This will lead us to the replica limit n—>1 of the CP^{n-1} sigma model, which at n>1 describes (2+1)-dimensional quantum antiferromagnets with SU(n) symmetry. I will then discuss a class of 3D loop models which (as well as being interesting polymer-like problems in their own right) provide a unifying description for line defects and SU(n) magnets, and an ideal platform for Monte Carlo simulations. I will describe our numerical results on such models, including preliminary results for the the ‘deconfined’ critical point much discussed in relation to the spin-1/2 square lattice antiferromagnet.