Abstract For short-ranged disordered quantum models in one dimension, the many-body-localization is analyzed via the adaptation to the many-body context (Serbyn et al 2015 Phys. Rev . X 5 041047) of the Thouless point of view on the Anderson transition: the question is whether a local interaction between two long chains is able to reshuffle completely the eigenstates (delocalized phase with a volume-law entanglement) or whether the hybridization between tensor states remains limited (many-body-localized phase with an area-law entanglement). The central object is thus the level of hybridization induced by the matrix elements of local operators, as compared with the difference of diagonal energies. The multifractal analysis of these matrix elements of local operators is used to analyze the corresponding statistics of resonances. Our main conclusion is that the critical point is characterized by the strong-multifractality spectrum f ( 0 ⩽ α ⩽ 2 ) = α 2 , well known in the context of Anderson localization in spaces of effective infinite dimensionality, where the size of the Hilbert space grows exponentially with the volume. Finally, the possibility of a delocalized non-ergodic phase near criticality is discussed.
Abstract For short-ranged disordered quantum models in one dimension, the many-body-localization is analyzed via the adaptation to the many-body context (Serbyn et al 2015 Phys. Rev . X 5 041047) of the Thouless point of view on the Anderson transition: the question is whether a local interaction between two long chains is able to reshuffle completely the eigenstates (delocalized phase with a volume-law entanglement) or whether the hybridization between tensor states remains limited (many-body-localized phase with an area-law entanglement). The central object is thus the level of hybridization induced by the matrix elements of local operators, as compared with the difference of diagonal energies. The multifractal analysis of these matrix elements of local operators is used to analyze the corresponding statistics of resonances. Our main conclusion is that the critical point is characterized by the strong-multifractality spectrum f ( 0 ⩽ α ⩽ 2 ) = α 2 , well known in the context of Anderson localization in spaces of effective infinite dimensionality, where the size of the Hilbert space grows exponentially with the volume. Finally, the possibility of a delocalized non-ergodic phase near criticality is discussed.
For short-ranged disordered quantum models in one dimension, the Many-Body-Localization is analyzed via the adaptation to the Many-Body context [M. Serbyn, Z. Papic and D.A. Abanin, PRX 5, 041047 (2015)] of the Thouless point of view on the Anderson transition : the question is whether a local interaction between two long chains is able to reshuffle completely the eigenstates (Delocalized phase with a volume-law entanglement) or whether the hybridization between tensor states remains limited (Many-Body-Localized Phase with an area-law entanglement). The central object is thus the level of Hybridization induced by the matrix elements of local operators, as compared with the difference of diagonal energies. The multifractal analysis of these matrix elements of local operators is used to analyze the corresponding statistics of resonances. Our main conclusion is that the critical point is characterized by the Strong-Multifractality Spectrum f(0≤α≤2)=2α, well known in the context of Anderson Localization in spaces of effective infinite dimensionality, where the size of the Hilbert space grows exponentially with the volume. As a consequence, the level statistics at criticality is the closest possible to the Poisson statistics of the Localized phase, and the Delocalized phase near the critical point is not ergodic.
Thermal and many-body localized phases are separated by a dynamical phase transition of a new kind. We analyze the distribution of off-diagonal matrix elements of local operators across this transition in two different models of disordered spin chains. We show that the behavior of matrix elements can be used to characterize the breakdown of thermalization and to extract the many-body Thouless energy. We find that upon increasing the disorder strength the system enters a critical region around the many-body localization transition. The properties of the system in this region are: (i) the Thouless energy becomes smaller than the level spacing, (ii) the matrix elements show critical dependence on the energy difference, and (iii) the matrix elements, viewed as amplitudes of a fictitious wave function, exhibit strong multifractality. This critical region decreases with the system size, which we interpret as evidence for a diverging correlation length at the many-body localization transition. Our findings show that the correlation length becomes larger than the accessible system sizes in a broad range of disorder strength values and shed light on the critical behavior near the many-body localization transition.
The space of one-dimensional disordered interacting quantum models displaying a many-body localization (MBL) transition seems sufficiently rich to produce critical points with level statistics interpolating continuously between the Poisson statistics of the localized phase and the Wigner–Dyson statistics of the delocalized phase. In this paper, we consider the strong disorder limit of the MBL transition, where the level statistics at the MBL critical point is close to the Poisson statistics. We analyze a one-dimensional quantum spin model, in order to determine the statistical properties of the rare extensive resonances that are needed to destabilize the MBL phase. At criticality, we find that the entanglement entropy can grow with an exponent 0 < α < 1 anywhere between the area law α = 0 and the volume law α = 1 , as a function of the resonances properties, while the entanglement spectrum follows the strong multifractality statistics. In the MBL phase near criticality, we obtain the simple value ν = 1 for the correlation length exponent. Independently of the strong disorder limit, we explain why, for the many-body localization transition concerning individual eigenstates, the correlation length exponent ν is not constrained by the usual Harris inequality ν ≥ 2 / d , so that there is no theoretical inconsistency with the best numerical measure ν = 0 . 8 ( 3 ) obtained by Luitz et al. (2015).
The interplay between quenched disorder and interaction effects opens the possibility in a closed quantum many-body system of a phase transition at finite energy density between an ergodic phase, which is governed by the laws of statistical physics, and a localized one, in which the degrees of freedom are frozen and ergodicity breaks down. The possible existence of a quantum phase transition at finite energy density is strongly questioning our understanding of the fundamental laws of nature and has generated an active field of research called many-body localization. This thesis consists of three parts and is dedicated to the understanding and characterization of the phenomenon of many-body localization, approaching it from complementary facets. In particular, borrowing methods and tools from different fields, we analyze timely problems. The first part of the thesis is devoted to detecting the many-body localization transition and to characterize both the ergodic and the localized phase it separates. Here we provide a characterization from two different perspectives: the first one is based on the study of local entanglement properties. In the second one, using tools from quantum-chaos theory, we attempt to answer the question of understanding time-irreversibility, and thus probing the breaking of ergodicity. We analyze experimentally viable observables. Moreover, we propose two different quantities to distinguish an Anderson insulating phase from a many-body localized one, which is one of the issues in experiments. The second part focuses on understanding the existence of a putative subdiffusive multifractal phase. Analyzing the quantum dynamics of the system in this region of the phase diagram, we point out the importance of finite-size effects, questioning the existence of this multifractal phase. We speculate with a possible scenario in which the diffusivity and thus ergodicity could be restored in the thermodynamic limit. Furthermore, we find that the propagation is highly non-Gaussian, which could have an important effect on understanding the critical point of the according transition. We tackle this problem also from a different angle. A possible toy-model to understand many-body localization entails the Anderson model on a random-regular graph. Also in the latter model the possible existence of an intermediate multifractal phase has been conjectured. There, studying the survival return probability of a particle with time, we give a new characterization of multifractal phases and give indication of the possible existence of this phase. Nevertheless, we also outline possible caveats. In the last part of this thesis we study the interplay between symmetry and correlated disorder in a non-interacting fermionic system. We show another possible mechanism for breaking localization. In particular, we focus on studying information and particle transport, emphasizing how the two types of propagation can be different.
Statistical analysis of the eigenfunctions of the Anderson tight-binding model with on-site disorder on regular random graphs strongly suggests that the extended states are multifractal at any finite disorder. The spectrum of fractal dimensions f(α) defined in Eq. (3) remains positive for α noticeably far from 1 even when the disorder is several times weaker than the one which leads to the Anderson localization; i.e., the ergodicity can be reached only in the absence of disorder. The one-particle multifractality on the Bethe lattice signals on a possible inapplicability of the equipartition law to a generic many-body quantum system as long as it remains isolated.
In contrast with Anderson localization where a genuine localization is observed in real space, the many-body localization (MBL) problem is much less understood in Hilbert space, the support of the eigenstates. In this Letter, using exact diagonalization techniques we address the ergodicity properties in the underlying N-dimensional complex networks spanned by various computational bases for up to L=24 spin-1/2 particles (i.e., Hilbert space of size N≃2.7×10^{6}). We report fully ergodic eigenstates in the delocalized phase (irrespective of the computational basis), while the MBL regime features a generically (basis-dependent) multifractal behavior, delocalized but nonergodic. The MBL transition is signaled by a nonuniversal jump of the multifractal dimensions.