Loschmidt Echo in the Time Crystal Phase of Disordered Spin Systems*

Motivated by recent studies on systems exhibiting discrete time translational symmetry breaking (DTTSB) we investigate the Loschmidt echo for disordered spin systems with special focus on the transverse Ising model. Starting from the $Z_{2}$ symmetric limit we analytically obtain the time evolution for various short-ranged correlated initial states. Our results enable us to draw an intuitive picture for the emergence of DTTSB, from which we generalize our findings to systems with $Z_{n}$ symmetry breaking.