Dissipative time crystals with long-range Lindbladians

Dissipative time crystals can appear in spin systems, when the Z2 symmetry of the Hamiltonian is broken by the environment, and the square of total spin operator S2 is conserved. In this paper, we relax the latter condition and show that time-translation-symmetry-breaking collective oscillations persist, in the thermodynamic limit, even in the absence of spin symmetry. We engineer an ad hoc Lindbladian using power-law-decaying spin operators and show that time-translation-symmetry breaking appears when the decay exponent obeys 0