We construct a complete set of local integrals of motion that characterize the many-body localized (MBL) phase. Our approach relies on the assumption that local perturbations act locally on the eigenstates in the MBL phase, which is supported by numerical simulations of the random-field XXZ spin chain. We describe the structure of the eigenstates in the MBL phase and discuss the implications of local conservation laws for its nonequilibrium quantum dynamics. We argue that the many-body localization can be used to protect coherence in the system by suppressing relaxation between eigenstates with different local integrals of motion.

Certain wave functions of non-interacting quantum chaotic systems can exhibit "scars" in the fabric of their real-space density profile. Quantum scarred wave functions concentrate in the vicinity of unstable periodic classical trajectories. We introduce the notion of many-body quantum scars which reflect the existence of a subset of special many-body eigenstates concentrated in certain parts of the Hilbert space. We demonstrate the existence of scars in the Fibonacci chain-the onedimensional model with a constrained local Hilbert space realized in the 51 Rydberg atom quantum simulator [H. Bernien et al., arXiv:1707.04344]. The quantum scarred eigenstates are embedded throughout the thermalizing many-body spectrum, but surprisingly lead to direct experimental signatures such as robust oscillations following a quench from a charge-density wave state found in experiment. We develop a model based on a single particle hopping on the Hilbert space graph, which quantitatively captures the scarred wave functions up to large systems of L = 32 atoms. Our results suggest that scarred many-body bands give rise to a new universality class of quantum dynamics, which opens up opportunities for creating and manipulating novel states with long-lived coherence in systems that are now amenable to experimental study.

Recent numerical work by Bardarson et. al.[1] revealed a slow, logarithmic in time, growth of the entanglement entropy for initial product states in a putative many-body localized phase. We show that this surprising phenomenon results from the dephasing due to exponentially small interactioninduced corrections to the eigenenergies of different states. For weak interactions, we find that the entanglement entropy grows as ξ ln(V t/ ), where V is the interaction strength, and ξ is the single-particle localization length. The saturated value of the entanglement entropy at long times is determined by the participation ratios of the initial state over the eigenstates of the subsystem. Our work shows that the logarithmic entanglement growth is a universal phenomenon characteristic of the many-body localized phase in any number of spatial dimensions, and reveals a broad hierarchy of dephasing time scales present in such a phase. Introduction. While it is well-known that arbitrarily weak disorder localizes all single-particle quantummechanical states in 1 and 2 dimensions, the effect of disorder potential on the states of interacting systems largely remains an open problem. Works [2,3] conjectured that localization in a many-body system survives in the presence of weak interactions. When the strength of the interactions is increased, at some critical value a transition to the delocalized phase -a "many-body localization" transition -takes place, as observed in the numerical simulations [4][5][6][7][8][9][10][11][12][13][14].

We consider disordered many-body systems with periodic time-dependent Hamiltonians in one spatial dimension. By studying the properties of the Floquet eigenstates, we identify two distinct phases: (i) a many-body localized (MBL) phase, in which almost all eigenstates have area-law entanglement entropy, and the eigenstate thermalization hypothesis (ETH) is violated, and (ii) a delocalized phase, in which eigenstates have volume-law entanglement and obey the ETH. The MBL phase exhibits logarithmic in time growth of entanglement entropy when the system is initially prepared in a product state, which distinguishes it from the delocalized phase. We propose an effective model of the MBL phase in terms of an extensive number of emergent local integrals of motion, which naturally explains the spectral and dynamical properties of this phase. Numerical data, obtained by exact diagonalization and time-evolving block decimation methods, suggest a direct transition between the two phases.

Recent realization of a kinetically-constrained chain of Rydberg atoms by Bernien et al. [Nature 551, 579 (2017)] resulted in the observation of unusual revivals in the many-body quantum dynamics. In our previous work, Turner et al. [arXiv:1711.03528], such dynamics was attributed to the existence of "quantum scarred" eigenstates in the many-body spectrum of the experimentally realized model. Here we present a detailed study of the eigenstate properties of the same model. We find that the majority of the eigenstates exhibit anomalous thermalization: the observable expectation values converge to their Gibbs ensemble values, but parametrically slower compared to the predictions of the eigenstate thermalization hypothesis (ETH). Amidst the thermalizing spectrum, we identify non-ergodic eigenstates that strongly violate the ETH, whose number grows polynomially with system size. Previously, the same eigenstates were identified via large overlaps with certain product states, and were used to explain the revivals observed in experiment. Here we find that these eigenstates, in addition to highly atypical expectation values of local observables, also exhibit sub-thermal entanglement entropy that scales logarithmically with the system size. Moreover, we identify an additional class of quantum scarred eigenstates, and discuss their manifestations in the dynamics starting from initial product states. We use forward scattering approximation to describe the structure and physical properties of quantum-scarred eigenstates. Finally, we discuss the stability of quantum scars to various perturbations. We observe that quantum scars remain robust when the introduced perturbation is compatible with the forward scattering approximation. In contrast, the perturbations which most efficiently destroy quantum scars also lead to the restoration of "canonical" thermalization. arXiv:1806.10933v2 [cond-mat.quant-gas]

We study dynamics of isolated quantum many-body systems whose Hamiltonian is switched between two different operators periodically in time. The eigenvalue problem of the associated Floquet operator maps onto an effective hopping problem. Using the effective model, we establish conditions on the spectral properties of the two Hamiltonians for the system to localize in energy space.We find that ergodic systems always delocalize in energy space and heat up to infinite temperature, for both local and global driving. In contrast, manybody localized systems with quenched disorder remain localized at finite energy.We support our conclusions by numerical simulations of disordered spin chains.We argue that our results hold for general driving protocols, and discuss their experimental implications.

Motivated by recent experimental observations of coherent many-body revivals in a constrained Rydberg atom chain, we construct a weak quasi-local deformation of the Rydberg blockade Hamiltonian, which makes the revivals virtually perfect. Our analysis suggests the existence of an underlying non-integrable Hamiltonian which supports an emergent SU(2)-spin dynamics within a small subspace of the many-body Hilbert space. We show that such perfect dynamics necessitates the existence of atypical, nonergodic energy eigenstates -quantum many-body scars. Furthermore, using these insights, we construct a toy model that hosts exact quantum many-body scars, providing an intuitive explanation of their origin. Our results offer specific routes to enhancing coherent many-body revivals, and provide a step towards establishing the stability of quantum many-body scars in the thermodynamic limit.Remarkable experimental advances have recently enabled studies of nonequilibrium dynamics of isolated, strongly interacting quantum systems [1][2][3]. In such systems, it is commonly believed that a generic state initialized far from equilibrium eventually thermalizes, whereupon any initial local information becomes unrecoverable [4][5][6]. While this process of thermalization provides the basis of statistical mechanics, it also poses challenges for building large-scale quantum devices. Hence, it is of fundamental interest to understand mechanisms to evade thermalization. Two well-studied possibilities include many-body localization in strongly disordered systems, and fine-tuned integrable systems [7][8][9].Recently, quench experiments with Rydberg atom arrays [10][11][12] have discovered non-thermalizing dynamics of a new kind [12]. Initialized in a high-energy Néel state, the system exhibited unexpectedly long-lived, periodic revivals, failing to thermalize on experimentally accessible timescales; in contrast, other high-energy product states exhibited thermalizing dynamics consistent with conventional expectations. arXiv:1812.05561v1 [quant-ph] 13 Dec 2018 * S. C. and C. J. T contributed equally to this work. arXiv:1812.05561v1 [quant-ph]

This article is a brief introduction to the rapidly evolving field of many‐body localization. Rather than giving an in‐depth review of the subject, our aspiration here is simply to introduce the problem and its general context, outlining a few directions where notable progress has been achieved in recent years. We hope that this will prepare the readers for the more specialized articles appearing in this dedicated Volume of Annalen der Physik, where these developments are discussed in more detail.