Statistical mechanics of vortices from field theory

We study with lattice Monte Carlo simulations the interactions and macroscopic behavior of a large number of vortices in the three-dimensional U(1) gauge + Higgs field theory, in an external magnetic field. We determine non-perturbatively the (attractive or repelling) interaction energy between two or more vortices, as well as the critical field strength Hc, the thermodynamical discontinuities, and the surface tension related to the boundary between the Meissner phase and the Coulomb phase in the type I region. We also investigate the emergence of vortex lattice and vortex liquid phases in the type II region. For the type I region the results obtained are in qualitative agreement with mean field theory, except for small values of Hc, while in the type II region there are significant discrepancies. These findings are relevant for superconductors and some models of cosmic strings, as well as for the electroweak phase transition in a magnetic field