Stable time rondeau crystals in dissipative many-body systems

Driven systems offer the potential to realize a wide range of nonequilibrium phenomena that are inaccessible in static systems, such as discrete time crystals. Time rondeau crystals with a partial temporal order have been proposed as a distinctive prethermal phase of matter in systems driven by structured random protocols. Yet, heating is inevitable in closed systems and time rondeau crystals eventually melt. We introduce dissipation to counteract heating and demonstrate stable time rondeau crystals, which persist indefinitely, in a many-body interacting system. A key ingredient is synchronization in the noninteracting limit, which allows for stable time rondeau order without generating excessive heating. The presence of many-body interaction competes with synchronization and a desynchronization phase transition occurs at a finite interaction strength. This transition is well captured via a linear stability analysis of the underlying stochastic processes.