  We exactly diagonalize the finite-size XY model with periodic boundary
conditions and analytically determine the ground state energy. We show that
there are two types of fermions: singles and pairs, whose dispersion relations
have a completely arbitrary gauge-dependent sign. It follows that the ground
state is determined by a competition between the vacuum states (with a suitable
gauge) of two parity sectors. We finally exhibit some points in finite size
systems that forerun criticality. They are associated to single Bogoliubov
fermions and to the level crossings between physical and unphysical states. In
the thermodynamic limit they approach the ground state and build up
singularities at logarithmic rates.
