Discrete Time Crystal

Floquet-driven Ising chain — spontaneous period-doubling

A discrete time crystal (DTC) in a periodically driven Ising chain spontaneously breaks the discrete time-translation symmetry T → 2T (period-doubling). The drive: step 1 applies π-pulse to all spins (±(π+ε)), step 2 applies random disorder fields. MBL (many-body localization) stabilizes the DTC phase against thermalization. Signature: magnetization ⟨Z(t)⟩ oscillates at half the drive frequency, robust to small ε.