We present a large scale exact diagonalization study of the one dimensional spin 1/2 Heisenberg model in a random magnetic field. In order to access properties at varying energy densities across the entire spectrum for system sizes up to L = 22 spins, we use a spectral transformation which can be applied in a massively parallel fashion. Our results allow for an energy-resolved interpretation of the many body localization transition including the existence of an extensive many-body mobility edge. The ergodic phase is well characterized by Gaussian orthogonal ensemble statistics, volumelaw entanglement, and a full delocalization in the Hilbert space. Conversely, the localized regime displays Poisson statistics, area-law entanglement and non ergodicity in the Hilbert space where a true localization never occurs. We perform finite size scaling to extract the critical edge and exponent of the localization length divergence. The interplay of disorder and interactions in quantum systems can lead to several intriguing phenomena, amongst which the so-called many-body localization has attracted a huge interest in recent years. Following precursors works [1][2][3][4], perturbative calculations [5,6] have established that the celebrated Anderson localization [7] can survive interactions, and that for large enough disorder, many-body eigenstates can also "localize" (in a sense to be precised later) and form a new phase of matter commonly referred to as the many-body localized (MBL) phase.The enormous boost of interest for this topic over the last years can probably be ascribed to the fact that the MBL phase challenges the very foundations of quantum statistical physics, leading to striking theoretical and experimental consequences [8,9]. Several key features of the MBL phase can be highlighted as follows. It is nonergodic, and breaks the eigenstate thermalization hypothesis (ETH) [10][11][12]: a closed system in the MBL phase does not thermalize solely following its own dynamics. The possible presence of a many-body mobility edge (at a finite energy density in the spectrum) indicates that conductivity should vanish in a finite temperature range in a MBL system [5,6]. Coupling to an external bath will eventually destroy the properties of the MBL phase, but recent arguments show that it can survive and be detected using spectral signatures for weak bath-coupling [13]. This leads to the suggestion that the MBL phase can be characterized experimentally, using e.g. controlled echo experiments on reasonably well-isolated systems with dipolar interactions [14][15][16][17]. Another appealing aspect (with experimental consequences for self-correcting memories) is that MBL systems can sustain long-range, possibly topological, order in situations where equilibrated systems would not [18][19][20][21][22]. Finally, a striking phenomenological approach [23] pinpoints that the MBL phase shares properties with integrable systems, with extensive local integrals of mo- ∈ {14, 15, 16, 17, 18, 19, 20, 22}. Red squares correspond to a visual e...

5 pages, 3 figuresInternational audienceMany-body localization is characterized by a slow logarithmic growth of the entanglement entropy after a global quantum quench while the local memory of an initial density imbalance remains at infinite time. We investigate how much the proximity of a many-body localized phase can influence the dynamics in the delocalized ergodic regime where thermalization is expected. Using an exact Krylov space technique, the out-of-equilibrium dynamics of the random-field Heisenberg chain is studied up to L=28 sites, starting from an initially unentangled high-energy product state. Within most of the delocalized phase, we find a sub-ballistic entanglement growth $S(t)\propto t^{1/z}$ with a disorder-dependent exponent $z\ge1$, in contrast with the pure ballistic growth $z=1$ of clean systems. At the same time, anomalous relaxation is also observed for the spin imbalance $I(t) \propto t^{-\zeta}$ with a continuously varying disorder-dependent exponent $\zeta$, vanishing at the transition. This provides a clear experimental signature for detecting this non-conventional regime