XY model on the circle: Diagonalization, spectrum, and forerunners of the quantum phase transition

We exactly diagonalize the finite-size XY model with periodic boundary conditions and analytically deter- mine the ground-state energy. We show that there are two types of Bogoliubov fermions, singles and pairs, whose dispersion relations have a completely arbitrary gauge-dependent sign. It follows that the ground state can be 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.