Absence of vortex condensation in a two dimensional fermionic XY model

Motivated by a puzzle in the study of two-dimensional lattice quantum electrodynamics with staggered fermions, we construct a two-dimensional fermionic model with a global U(1) symmetry. Our model can be mapped into a model of closed packed dimers and plaquettes. Although the model has the same symmetries as the XY model, we show numerically that the model lacks the well-known Kosterlitz-Thouless phase transition. The model is always in the gapless phase showing the absence of a phase with vortex condensation. In other words the low energy physics is described by a noncompact U(1) field theory. We show that by introducing an even number of layers one can introduce vortex condensation within the model and thus also induce a Kosterlitz-Thouless transition.