Abstract
 
Periodic driving of quantum systems provides a  natural experimental tool for probing and modifying their properties. In particular, periodic driving has been recently used to realize topologically non-trivial band structures in optical lattices. Yet, theoretically relatively little is known about the general properties of driven interacting systems.
     In this talk, I will present a theory of many-body localized, periodically driven many-body systems [1,2]. I will show that many-body localization persists under periodic driving at high enough driving frequency. The Floquet operator (evolution operator over one driving period) can be represented as an exponential of an effective time-independent Hamiltonian, which is a sum of local terms and is itself MBL. The Floquet eigenstates in this case have area-law entanglement entropy, and there exists an extensive set of local integrals of motion. I will further argue that at sufficiently low frequency, there is always delocalization, owing to a large number of many-body level crossings and non-diabatic Landau-Zener transition between them. I will propose a phase diagram of driven MBL systems. Our results provide new experimental signatures of many-body localization, and indicate breakdown of the linear-response theory and classic Mott formula for AC conductivity in localized systems. I will close by discussing periodically driven ergodic (delocalized) systems, and, in particular, will derive general strong upper bounds on their heating rates. 
 
[1] D. A. Abanin, W. De Roeck, F. Huveneers, "A theory of many-body localization in periodically driven systems", arXiv:1412.4752 (2014). 
[2] P. Ponte, Z. Papic, F. Huveneers, D. A. Abanin, "Many-body localization in periodically driven systems", arXiv:1410.8518 (2014), PRL, in press.