  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 non-compact 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 KT transition.
