Rigidity and Criticality in Discrete Time Crystals

Norman Y. Yao

Koki Chinzei, Tatsuhiko N. Ikeda

We consider a dissipative quantum Ising model periodically driven by a train of $\ensuremath{\pi}$ pulses and investigate dissipative discrete time crystals (DTCs) in solids. In this model, the interaction between the spins spontaneously breaks the discrete time translation symmetry, giving rise to a dissipative DTC, where two ferromagnetic states are switched alternately by each pulse. We microscopically describe the generic dissipation due to thermal contact to an equilibrium heat bath using the Bloch-Redfield equation. In contrast to other DTC studies, this dissipation stabilizes, rather than destroys, the DTC order without fine-tuning as long as the temperature is low enough. Invoking the time-dependent mean-field theory and solving self-consistently the periodic drive, dissipation, and DTC order parameter, we investigate the nonequilibrium DTC phase transition and determine the critical exponents, including a dynamical one. We also find phase transitions without an equilibrium counterpart: a nontrivial interplay of the periodic drive and dissipation gives rise to reentrant DTC transition when changing the pulse interval at a fixed temperature. Also, to demonstrate the rigidity of the DTC, we consider imperfect $\ensuremath{\pi}$ pulses, showing that the DTC is robust against the small imperfections and finding that discrete time quasicrystals can appear for the larger imperfections. Together with experimental proposals in magnetic materials, our results pave the way for realizing the DTC and for uncovering nonequilibrium critical phenomena in real solid-state materials.

Krzysztof Sacha

Norman Y. Yao, Andrew C. Potter, Ionut-Dragos Potirniche +1 more

Despite being forbidden in equilibrium, spontaneous breaking of time translation symmetry can occur in periodically driven, Floquet systems with discrete time-translation symmetry. The period of the resulting discrete time crystal is quantized to an integer multiple of the drive period, arising from a combination of collective synchronization and many body localization. Here, we consider a simple model for a one-dimensional discrete time crystal which explicitly reveals the rigidity of the emergent oscillations as the drive is varied. We numerically map out its phase diagram and compute the properties of the dynamical phase transition where the time crystal melts into a trivial Floquet insulator. Moreover, we demonstrate that the model can be realized with current experimental technologies and propose a blueprint based upon a one dimensional chain of trapped ions. Using experimental parameters (featuring long-range interactions), we identify the phase boundaries of the ion-time-crystal and propose a measurable signature of the symmetry breaking phase transition.

Norman Y. Yao

Norman Y. Yao, Chetan Nayak, Leon Balents +1 more