Band structure engineering and non-equilibrium dynamics in Floquet topological insulators

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 begins with the key principles underlying Floquet band engineering, wherein such fields are used to change the topological properties of a system's single-particle spectrum. In contrast to equilibrium systems, non-trivial band structure topology in a driven many-body system does not guarantee that robust topological behaviour will be observed. In particular, periodically driven many-body systems tend to absorb energy from their driving fields and thereby tend to heat up. We survey various strategies for overcoming this challenge of heating and for obtaining new topological phenomena in this non-equilibrium setting. We describe how drive-induced topological edge states can be probed in the regime of mesoscopic transport, and three routes for observing topological phenomena beyond the mesoscopic regime: long-lived transient dynamics and prethermalization, disorder-induced many-body localization, and engineered couplings to external baths. We discuss the types of phenomena that can be explored in each of the regimes covered, and their experimental realizations in solid-state, cold atomic, and photonic systems. Time-periodic fields provide a versatile platform for inducing non-equilibrium topological phenomena in quantum systems. We discuss how such fields can be used for topological band structure engineering, and the conditions for observing robust topological behaviour in a many-body setting. Key pointsTime-periodic fields provide a versatile platform for inducing non-equilibrium topological phenomena in quantum systems.In contrast to equilibrium systems, non-trivial band structure topology does not guarantee that robust topological behaviour will be observed in a many-body system.Various strategies can be employed to obtain novel topological phenomena in this non-equilibrium many-body setting.Driving-induced topological edge states can be revealed through (non-quantized) mesoscopic transport.Working in regimes where heating rates are strongly suppressed allows robust topological behaviour to be observed in long-lived transients.Many-body localization due to strong disorder provides a mechanism for completely eliminating heating in certain types of systems, allowing a sharp delineation of intrinsic phases of many-body Floquet systems.Bath engineering provides a powerful means for ensuring that the topological properties of a driven system coupled to its natural environment are reflected in its steady state.