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

TL;DR

This paper provides an exact diagonalization of the finite-size XY model on a circle, revealing the structure of its fermionic excitations, the competition between parity sectors, and precursors to quantum criticality.

Contribution

It analytically determines the ground state energy and characterizes the fermionic spectrum, including gauge-dependent signs and finite-size precursors to phase transitions.

Findings

Exact diagonalization of the XY model spectrum.

Identification of two types of fermions with gauge-dependent signs.

Finite-size points indicating approaching criticality.

Abstract

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.