Symmetries and conserved quantities of boundary time crystals in generalized spin models

We investigate how symmetries and conserved quantities relate to the occurrence of boundary time crystal (BTC) phase in a generalized spin model with Lindblad dissipation. BTCs are nonequilibrium matter which system, coupled an external environment, breaks continuous translational invariance. perform detailed mean-field study aided by finite-size analysis quantum $p,q$-spin-interaction $p$-spin-interaction can be implemented fully connected spin-1/2 ensembles. find following conditions for observation BTC phase. First, appears when discrete symmetry held Hamiltonian, ${\mathbb{Z}}_{2}$ considered models, is explicitly broken jump operators. Second, system must uniformly same bath order preserve total angular momentum during evolution. If these not satisfied, any oscillatory behavior only as transient dynamics time-independent stationary state eventually reached. Our results suggest that two elements may general requirements stable relating arbitrary models.