Prethermalization refers to the transient phenomenon where a system thermalizes according to a Hamiltonian that is not the generator of its evolution. We provide here a rigorous framework for quantum spin systems where prethermalization is exhibited for very long times. First, we consider quantum spin systems under periodic driving at high frequency ν. We prove that up to a quasi-exponential time τ * ∼ e c ν log 3 ν , the system barely absorbs energy. Instead, there is an effective local Hamiltonian D that governs the time evolution up to τ * , and hence this effective Hamiltonian is a conserved quantity up to τ * . Next, we consider systems without driving, but with a separation of energy scales in the Hamiltonian. A prime example is the Fermi-Hubbard model where the interaction U is much larger than the hopping J. Also here we prove the emergence of an effective conserved quantity, different from the Hamiltonian, up to a time τ * that is (almost) exponential in U/J.

We propose a theory that describes quantitatively the (in)stability of fully MBL systems due to ergodic, i.e. delocalized, grains, that can be for example due to disorder fluctuations. The theory is based on the ETH hypothesis and elementary notions of perturbation theory. The main idea is that we assume as much chaoticity as is consistent with conservation laws. The theory describes correctly -even without relying on the theory of local integrals of motion (LIOM)-the MBL phase in 1 dimension at strong disorder. It yields an explicit and quantitative picture of the spatial boundary between localized and ergodic systems. We provide numerical evidence for this picture.When the theory is taken to its extreme logical consequences, it predicts that the MBL phase is destabilised in the long time limit whenever 1) interactions decay slower than exponentially in d = 1 and 2) always in d > 1. Finer numerics is required to assess these predictions.

We derive general bounds on the linear response energy absorption rates of periodically driven many-body systems of spins or fermions on a lattice. We show that for systems with local interactions, energy absorption rate decays exponentially as a function of driving frequency in any number of spatial dimensions. These results imply that topological many-body states in periodically driven systems, although generally metastable, can have very long lifetimes. We discuss applications to other problems, including decay of highly energetic excitations in cold atomic and solid-state systems.

We establish some general dynamical properties of quantum many-body systems that are subject to a high-frequency periodic driving. We prove that such systems have a quasi-conserved extensive quantity H * , which plays the role of an effective static Hamiltonian. The dynamics of the system (e.g., evolution of any local observable) is well-approximated by the evolution with the Hamiltonian H * up to time τ * , which is exponentially large in the driving frequency. We further show that the energy absorption rate is exponentially small in the driving frequency. In cases where H * is ergodic, the driven system prethermalizes to a thermal state described by H * at intermediate times t τ * , eventually heating up to an infinite-temperature state after times t ∼ τ * . Our results indicate that rapidly driven many-body systems generically exhibit prethermalization and very slow heating. We briefly discuss implications for experiments which realize topological states by periodic driving.

We present a theory of periodically driven, many-body localized (MBL) systems. We argue that MBL 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 quasi-local terms and is itself fully MBL. We derive this result by constructing a sequence of canonical transformations to remove the time-dependence from the original Hamiltonian. When the driving evolves smoothly in time, the theory can be sharpened by estimating the probability of adiabatic Landau-Zener transitions at many-body level crossings. In all cases, we argue that there is delocalization at sufficiently low frequency. We propose a phase diagram of driven MBL systems.Introduction. Recently, there has been much interest in quantum many-body localized (MBL) systems and their properties [1][2][3][4][5][6][7][8][9][10][11][12][13][14][17][18][19][20][21][22][23][24]. MBL phase is characterized by an extensive set of emergent local integrals of motion (LIOMs) [12,13], which lead to quantum ergodicity breaking, and in particular, absence of thermalization. Therefore, MBL systems cannot be described by conventional statistical mechanics. Existing works explored experimental manifestations of MBL systems, and predicted universal dynamical properties following a sudden quantum quench, including logarithmic growth of entanglement entropy [6, 8-10, 12, 13], as well as characteristic decay [22] and revivals [23] of local observables.In this paper, we study the behaviour of MBL systems under periodic driving. Previous works on driven many-body systems focused mostly on the translationally invariant case [25][26][27][28][29][30]. In particular, D'Alessio and Polkovnikov [27] conjectured that, if the dynamics is generated by switching between an ergodic and an integrable (but translationally-invariant) Hamiltonian, a transition will be observed in function of the driving frequency: At low frequency, the system shows heating to an infinite temperature, while at high frequency the dynamics is described by an effective Hamiltonian written as a sum of local terms, leading to localization in the energy space. Though very long time scales can indeed be needed for energy to get dissipated [31], it was argued that driven ergodic systems typically delocalize and heat up to an infinite temperature at any driving frequency [28,30,32].There are three main motivations to our work. First, studying the response of many-body systems to periodically varying fields is a conventional experimental probe in systems of cold atoms in optical lattices [33,34], which are promising candidates for realizing the MBL phase [35,36]. Second, theoretically little is known about general properties of quantum many-body systems under time-varying fields (beyond linear-response). Finally, we investigate the conjecture in [27], in the context of MBL systems, where counter-arguments based on ergodicity fail in an obvious way.We c...