Long-Lived Interacting Phases of Matter Protected by Multiple Time-Translation Symmetries in Quasiperiodically Driven Systems

We show how a large family of interacting nonequilibrium phases of matter can arise from the presence of multiple time-translation symmetries, which occur by quasiperiodically driving an isolated, quantum many-body system with two or more incommensurate frequencies. These phases are fundamentally different from those realizable in time-independent or periodically driven (Floquet) settings. Focusing on high-frequency drives with smooth time dependence, we rigorously establish general conditions for which these phases are stable in a parametrically long-lived preheating regime. We develop a formalism to analyze the effect of the multiple time-translation symmetries on the dynamics of the system, which we use to classify and construct explicit examples of the emergent phases. In particular, we discuss time quasicrystals which spontaneously break the time-translation symmetries, as well as time-translation symmetry-protected topological phases.