Geometrical Origin of Tricritical Points of various U(1) Lattice Models

Abstract

We review the dual relationship between various compact U(1) lattice models and Abelian Higgs models, the latter being the disorder field theories of line-like topological excitations in the systems. We point out that the predicted first-order transitions in the Abelian Higgs models (Coleman-Weinberg mechanism) are, in three dimensions, in contradiction with direct numerical investigations in the compact U(1) formulation since these yield continuous transitions in the major part of the phase diagram. In four dimensions, there are indications from Monte Carlo data for a similar situation. Concentrating on the strong-coupling expansion in terms of geometrical objects, surfaces or lines, with certain statistical weights, we present semi-quantitative arguments explaining the observed cross-over from first-order to continuous transitions by the balance between the lowest two weights (``2:1 ratio'') of these geometrical objects.