Coexistence of distinct Discrete Time-Crystalline orders in the Floquet Lipkin-Meshkov-Glick model

We examine the distinct discrete time crystals (DTCs) that emerge in the Lipkin-Meshkov-Glick model, subjected to spatially nonuniform periodic driving. Intriguingly, we demonstrate that by appropriately tailoring the drive protocol, distinct DTC orders can be realized in different spatial regions of the system. Consequently, the system exhibits spatially varying sub-harmonic responses with distinct frequencies. We employ a semi-classical analysis to establish the stability of these co-existing DTC orders in the thermodynamic limit. Furthermore, we establish the stability of the stability of these co-existing DTCs in the presence of quantum fluctuations. Our results establish spatially structured driving as a powerful route to engineer novel forms of time-crystalline order.