Non-equilibrium topological phenomena can be induced in quantum many-body systems
using time-periodic fields (for example, by laser or microwave illumination). This Review …

There have been significant recent advances in realizing band structures with geometrical
and topological features in experiments on cold atomic gases. This review summarizes …

Time crystals are time-periodic self-organized structures postulated by Frank Wilczek in
2012. While the original concept was strongly criticized, it stimulated at the same time an …

Despite being forbidden in equilibrium, spontaneous breaking of time translation symmetry
can occur in periodically driven, Floquet systems with discrete time-translation symmetry …

Clean and interacting periodically driven systems are believed to exhibit a single, trivial
“infinite-temperature” Floquet-ergodic phase. In contrast, here we show that their disordered …

Experimental advances have allowed for the exploration of nearly isolated quantum many-
body systems whose coupling to an external bath is very weak. A particularly interesting …

Experimental progress in atomic, molecular, and optical platforms in the last decade has
stimulated strong and broad interest in the quantum coherent dynamics of many long-range …

Floquet systems are governed by periodic, time-dependent Hamiltonians. Prima facie they
should absorb energy from the external drives involved in modulating their couplings and …

We define what it means for time translation symmetry to be spontaneously broken in a
quantum system and show with analytical arguments and numerical simulations that this …

Recent work has shown that a variety of novel phases of matter arise in periodically driven
Floquet systems. Among these are many-body localized phases which spontaneously break …