Prethermal Phases of Matter Protected by Time-Translation Symmetry

In a periodically driven (Floquet) system, there is the possibility for new phases of matter, not present in stationary systems, protected by discrete time-translation symmetry. This includes topological phases protected in part by time-translation symmetry, as well as phases distinguished by the spontaneous breaking of this symmetry, dubbed “Floquet time crystals.” We show that such phases of matter can exist in the prethermal regime of periodically driven systems, which exists generically for sufficiently large drive frequency, thereby eliminating the need for integrability or strong quenched disorder, which limited previous constructions. We prove a theorem that states that such a prethermal regime persists until times that are nearly exponentially long in the ratio of certain couplings to the drive frequency. By similar techniques, we can also construct stationary systems that spontaneously break continuous time-translation symmetry. Furthermore, we argue that for driven systems coupled to a cold bath, the prethermal regime could potentially persist to infinite time.

Popular Summary

Many phases of matter exist. Some are mundane such as solids and liquids, and some are exotic such as Bose-Einstein condensates, but all are defined by distinct differences in their physical properties. For example, a substance can have either a regular atomic crystal structure (solid) or not (liquid)—there are no in-betweens. Phases of matter are usually defined in thermal equilibrium, which means that the system does not exchange heat with its surroundings. The study of phases away from equilibrium, for example, in systems exposed to a time-varying field such as a regularly oscillating magnetic field, is largely uncharted territory. We have developed a mathematical formulation that shows that new phases of matter can exist in such systems.

Previous proposals for similar phases of matter have required fine-tuning or decoupling from the surrounding environment. We invoke the idea of “prethermalization”: A system can absorb energy so slowly that its energy does not look like it is changing until a very long time has passed. Our scenario, which shows that these prethermal states can exhibit unique phases of matter, is more robust to interactions with the environment than previous models. The sharp difference between phases lies in the nature of the micromotions, or the way the system oscillates when observed at small time intervals. An example is a “Floquet time crystal,” which oscillates at a different frequency than the applied field. Without an oscillating field, the system would not exhibit any micromotions, so these are entirely new and unconventional phases of matter.

These phases exist only while the system is prethermal. Future work could focus on how a cold bath might counteract heating that ends the prethermal state, possibly extending the duration of such phases to infinite time.

Figure 1

The expected time dependence of $⟨{\sigma }_i^z⟩$ at stroboscopic times, starting from a low-temperature state with respect to $\mathcal{U}D{\mathcal{U}}^{†}$ (for example, for a state with all spins polarized in the $z$ direction.). Panel (a) shows the prethermal time-crystal phase, and panel (b) shows the non-time-crystal prethermal phase.

Figure 2

So long as the energy inflow due to noise and periodic driving is balanced by the outflow to a cold thermal bath, giving a low-energy steady state, oscillations at a fraction of the drive frequency will be observed.