PACS 74.62.En -Effects of disorder PACS 64.60.an -Finite-size systems (General studies of phase transitions) PACS 72.15.Rn -Localization effects (Anderson or weak localization)Abstract -We present a detailed analysis of the length-and timescales needed to approach the critical region of MBL from the delocalised phase, studying both eigenstates and the time evolution of an initial state. For the eigenstates we show that in the delocalised region there is a single length, which is a function of disorder strength, controlling the finite-size flow. Small systems look localised, and only for larger systems do resonances develop which restore ergodicity in the form of the eigenstate thermalisation hypothesis. For the transport properties, we study the time necessary to transport a single spin across a domain wall, showing how this grows quickly with increasing disorder, and compare it with the Heisenberg time. For a sufficiently large system the Heisenberg time is always larger than the transport time, but for a smaller system this is not necessarily the case. We conclude that the properties of the MBL transition cannot be explored using the system sizes or times available to current numerical and experimental studies.

Previous studies revealed a crucial effect of symmetries on the properties of a single particle moving in a disorder potential. More recently, a phenomenon of many-body localization (MBL) has been attracting much theoretical and experimental interest. MBL systems are characterized by the emergence of quasi-local integrals of motion, and by the area-law entanglement entropy scaling of its eigenstates. In this paper, we investigate the effect of a non-Abelian SU (2) symmetry on the dynamical properties of a disordered Heisenberg chain. While SU (2) symmetry is inconsistent with the conventional MBL, a new non-ergodic regime is possible. In this regime, the eigenstates exhibit faster than area-law, but still a strongly sub-thermal scaling of entanglement entropy. Using extensive exact diagonalization simulations, we establish that this non-ergodic regime is indeed realized in the strongly disordered Heisenberg chains. We use real-space renormalization group (RSRG) to construct tree-tensor-network approximation to excited eigenstates, and demonstrate the accuracy of this procedure for systems of size up to L = 26. As the effective disorder strength is decreased, a crossover to the thermalizing phase occurs. To establish the ultimate fate of the nonergodic regime in the thermodynamic limit, we develop a novel approach for describing many-body processes that are usually neglected by RSRG. This approach is capable of describing systems of size L 2000. We characterize the resonances that arise due to such processes, finding that they involve an ever growing number of spins as the system size is increased. Crucially, the probability of finding resonances grows with the system's size. Even at strong disorder, we can identify a large lengthscale beyond which resonances proliferate. Presumably, this eventually would drive the system to a thermalizing phase. However, the extremely long thermalization time scales indicate that a broad non-ergodic regime will be observable experimentally. Our study demonstrates that, similar to the case of single-particle localization, symmetries control dynamical properties of disordered, many-body systems. The approach introduced here provides a versatile tool for describing a broad range of disordered many-body systems, well beyond sizes accessible in previous studies.arXiv :1902.09236v1 [cond-mat.str-el]

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