Non-equilibrium entanglement asymmetry for discrete groups: the example of the XY spin chain

Entanglement asymmetry is a novel quantity that, using entanglement methods, measures how much a symmetry is broken in a part of an extended quantum system. So far, it has only been used to characterise the breaking of continuous Abelian symmetries. In this paper, we extend the concept to cyclic ZN groups. As an application, we consider the XY spin chain, in which the ground state spontaneously breaks the Z2 spin parity symmetry in the ferromagnetic phase. We thoroughly investigate the non-equilibrium dynamics of this symmetry after a global quantum quench, generalising known results for the standard order parameter.