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<?<=1. This model shows a surprisingly rich phase diagram, including the time-crystal phase as well as first-order, second-order, and continuous transitions of the fixed points. We study the phase diagram and the magnetization dynamics in the mean-field approximation. We prove that this approximation is quantitatively accurate, when 0<?<1 and the thermodynamic limit is taken, because the system does not develop sizable quantum fluctuations, if the Gaussian approximation is considered. © 2022 American Physical Society.