Absence of Thermalization, Emergent Conservation laws, and the Statistical Mechanics of Periodically Driven Quantum Matter

When driven periodically in time, a closed quantum many-body
system is expected to absorb energy without bound, and get heated up. In
absence of any obvious conservation law, the system is consequently
expected to explore its Hilbert space ergodically. In the long run, this
corresponds to a locally "infinite T" like scenario for the system.
However, when the drive is strong enough, quantum interference can give
rise to a surprisingly different scenario -- perpetually stable
non-thermal states can appear as a consequence of the emergence of new
approximate conservation laws that are not present in the undriven
systems. This challenges the simple extrapolation of the fundamental
postulate of ergodicity of statistical mechanics to periodically driven
systems and opens up the possibility of stabilizing novel non-equilibrium
phases of interacting quantum matter in close systems. Here we will
discuss these developments in a self-contained way - from the beginning up
to the open front.