Cavity Induced Time Crystals in Spin Chains

We study the emergence of discrete time-crystalline order in a disordered Ising spin chain coupled to a quantized cavity field. In absence of the cavity, the system is governed by a monochromatically driven Hamiltonian, whose dynamics, expressed in the Majorana representation, exhibit robust period-doubled oscillations and characteristic finite-size scaling. A further mean-field analysis shows that the cavity field provides an effective dynamical feedback mechanism, reproducing the same oscillatory behavior observed in the Floquet regime. These results establish that the underlying spin system can host a stable time-crystalline phase robust to the disorder in the interaction. When the cavity is introduced, the drive is replaced by cavity dynamics in a time-independent Hamiltonian. Although a full treatment of dissipation effects remains an open question (to be addressed in future work), our current findings establish a theoretical framework that bridges driven and self-organized realizations of time crystals, which, in turn, highlights the cavity as a natural engine for persistent temporal order in quantum many-body systems.