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The entanglement asymmetry is a novel quantity that, using entanglementmethods, measures how much a symmetry is broken in a part of an extendedquantum system. So far it has only been used to characterise the breaking ofcontinuous Abelian symmetries. In this paper, we extend the concept to cyclic$\mathbb{Z}_N$ groups. As an application, we consider the XY spin chain, inwhich the ground state spontaneously breaks the $\mathbb{Z}_2$ spin paritysymmetry in the ferromagnetic phase. We thoroughly investigate thenon-equilibrium dynamics of this symmetry after a global quantum quench,generalising known results for the standard order parameter.

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