We experimentally observe many-body localization of interacting fermions in a one-dimensional quasi-random optical lattice. We identify the manybody localization transition through the relaxation dynamics of an initially-prepared charge density wave. For sufficiently weak disorder the time evolution appears ergodic and thermalizing, erasing all remnants of the initial order. In contrast, above a critical disorder strength a significant portion of the initial ordering persists, thereby serving as an effective order parameter for localization. The stationary density wave order and the critical disorder value show a distinctive dependence on the interaction strength, in agreement with numerical simulations. We connect this dependence to the ubiquitous logarithmic growth of entanglement entropy characterizing the generic many-body localized phase.Introduction The ergodic hypothesis is one of the central principles of statistical physics. In ergodic time evolution of a quantum many-body system, local degrees of freedom become fully entangled with the rest of the system, leading to an effectively classical hydrodynamic evolution of the remaining slow observables [1]. Hence, ergodicity is responsible for the demise of observable quantum correlations in the dynamics of large many-body systems and forms the basis for the emergence of local thermodynamic equilibrium in isolated quantum systems [2,3,4]. It is therefore of fundamental interest to investigate how ergodicity breaks down and search for alternative, genuinely quantum paradigms in the dynamics, and to understand the long-time stationary states that ensue in the absence of ergodicity.One path to breaking ergodicity is provided by the study of integrable models, where thermalization is prevented due to the constraints imposed on the dynamics by an infinite set of conservation rules. Such models have been realized and studied in a number of experiments with ultracold atomic gases [5,6,7]. However, integrable models represent very special and fine-tuned situations, making it difficult to extract general underlying principles.Theoretical studies over the last decade point to many-body localization (MBL) in a disordered isolated quantum system as a more generic alternative to thermalization dynamics. In his original paper on single-particle localization, Anderson already speculated that interacting many-body systems subject to sufficiently strong disorder would also fail to thermalize [8]. Only recently, however, have convincing theoretical arguments been put forward that Anderson localization remains stable under the addition of moderate interactions, even in highly excited many-body states [9,10,11]. Further theoretical studies have established the many-body localized state as a distinct dynamical phase of matter that exhibits novel universal behavior [12,13,14,15,16,17,18,19,20,21,22]. In particular, the relaxation of local observables does not follow the conventional paradigm of thermalization and is expected to show explicit breaking of ergodicity. In many ways, ...

We formulate a theory of the many-body localization transition based on a novel real-space renormalization group (RG) approach. The results of this theory are corroborated and intuitively explained with a phenomenological effective description of the critical point and of the "badly conducting" state found near the critical point on the delocalized side. The theory leads to the following sharp predictions: (i) The delocalized state established near the transition is a Griffiths phase, which exhibits subdiffusive transport of conserved quantities and sub-ballistic spreading of entanglement. The anomalous diffusion exponent α < 1=2 vanishes continuously at the critical point. The system does thermalize in this Griffiths phase. (ii) The many-body localization transition is controlled by a new kind of infinite-randomness RG fixed point, where the broadly distributed scaling variable is closely related to the eigenstate entanglement entropy. Dynamically, the entanglement grows as ∼ log t at the critical point, as it does in the localized phase. (iii) In the vicinity of the critical point, the ratio of the entanglement entropy to the thermal entropy and its variance (and, in fact, all moments) are scaling functions of L=ξ, where L is the length of the system and ξ is the correlation length, which has a power-law divergence at the critical point.

We formulate a dynamical real space renormalization group (RG) approach to describe the time evolution of a random spin-1/2 chain, or interacting fermions, initialized in a state with fixed particle positions. Within this approach we identify a many-body localized state of the chain as a dynamical infinite randomness fixed point. Near this fixed point our method becomes asymptotically exact, allowing analytic calculation of time dependent quantities. In particular, we explain the striking universal features in the growth of the entanglement seen in recent numerical simulations: unbounded logarithmic growth delayed by a time inversely proportional to the interaction strength. This is in striking contrast to the much slower entropy growth as loglogt found for noninteracting fermions with bond disorder. Nonetheless, even the interacting system does not thermalize in the long time limit. We attribute this to an infinite set of approximate integrals of motion revealed in the course of the RG flow, which become asymptotically exact conservation laws at the fixed point. Hence we identify the many-body localized state with an emergent generalized Gibbs ensemble.

We survey the recent progress made in understanding non equilibrium dynamics in closed random systems. The emphasis is on the important role played by concepts from quantum information theory and on the application of systematic renormalization group methods to capture universal aspects of the dynamics. Finally we outline some outstanding open questions, which includee the description of the many-body localization phase transition and identifying physical systems that will allow systematic experimental study of these phenomena.

Topological phases are characterized by edge states confined near the boundaries by a bulk energy gap. On raising temperature, these edge states are typically lost due to mobile thermal excitations. However, disorder can localize an isolated many-body system, potentially allowing for a sharply defined topological phase even in a highly excited state. We explicitly demonstrate this in a model of a disordered, one-dimensional magnet with spin one-half edge excitations. Furthermore, we show that the time evolution of a simple, highly excited state reveals quantum coherent edge spins. In particular, we demonstrate the coherent revival of an edge spin over a time scale that grows exponentially with system size. This is in sharp contrast to the general expectation that quantum bits strongly coupled with a hot many-body system will rapidly lose coherence. This result opens new directions in the study of topologically protected quantum dynamics.