We have studied various aspects of the critical properties of the Abelian Higgs model. The initial motivation to study this model is its relation to superconductivity, but the results extend beyond the realms of superconductivity. This thesis contains an introductory part and three research papers, all related to different aspects of the Abelian Higgs model.
Paper 1: We have investigated the properties of the model using a dual vortex representation. By focusing on the propagators of the gauge field A and the dual gauge field h we find a nice demonstration of the fact that the dual of a neutral condensate is isomorphic to a charged condensate. Finally this also provides firm support for the existence of a stable charged fixed point in the theory, distinct from the 3DXY fixed point.
Paper 2: The critical fluctuations in the Abelian Higgs model are vortex loops. We have studied the geometrical properties of these loops, and by using duality we have obtained scaling relations between the fractal dimension DH of the loops and the anomalous dimension ηφ of the dual field theory.
Paper 3: We have calculated the GL parameter κtri separating a first order metal to superconductor transition from a second order one, κtri =(0.76±0.04))/√2. We also argue qualitatively that this κtri is the value separating type-I and type-II behavior, in contrast to the conventional value 1=√2. The calculations have been done including fluctuations in the amplitude and the phase of the matter-field, as well as fluctuations in the gauge field.
Paper 4: We have determined the effective interaction between vortices in the Ginzburg-Landau model from large-scale Monte-Carlo simulations. We find a change, in the form of a crossover, from attractive to repulsive effective vortex interactions in an intermediate range of Ginzburg-Landau parameters κε[0.76; 1]=√2, depending on temperature. We present a simple physical picture of the crossover, and relate it to observations in Ta and Nb elemental superconductors which have low-temperature values of κ in the relevant range.