A central theoretical issue at the core of the current research on many-body localization (MBL) consists in characterizing the statistics of rare long-range resonances in many-body eigenstates. This is of paramount importance to understand: (i) the critical properties of the MBL transition and the mechanism for its destabilization through quantum avalanches; (ii) the unusual transport and anomalously slow out-of-equilibrium relaxation when the transition is approached from the metallic side. In order to study and characterize such long-range rare resonances, we develop a large-deviations approach based on an analogy with the physics of directed polymers in random media, and in particular with their freezing glass transition on infinite-dimensional graphs. The basic idea is to enlarge the parameter space by adding an auxiliary parameter (which plays the role of the inverse temperature in the directed polymer formulation) which allows us to fine-tune the effect of anomalously large outliers in the far-tails of the probability distributions of the transmission amplitudes between far-away many-body configurations in the Hilbert space. We first benchmark our approach onto two non-interacting paradigmatic toy models, namely the single-particle Anderson model on the (loop-less) Cayley tree and the Rosenzweig-Porter random matrix ensemble, and then apply it to the study of a class of disordered quantum spin chains in a transverse field. This analysis shows the existence of a broad disorder range in which rare, long-distance resonances, that may form only for a few specific realizations of the disorder and a few specific choice of the random initial state, destabilize the MBL phase, while the genuine MBL transition is shifted to much larger values of the disorder than originally thought.
A central theoretical issue at the core of the current research on many-body localization (MBL) consists in characterizing the statistics of rare long-range resonances in many-body eigenstates. This is of paramount importance to understand: (i) the critical properties of the MBL transition and the mechanism for its destabilization through quantum avalanches; (ii) the unusual transport and anomalously slow out-of-equilibrium relaxation when the transition is approached from the metallic side. In order to study and characterize such long-range rare resonances, we develop a large-deviations approach based on an analogy with the physics of directed polymers in random media, and in particular with their freezing glass transition on infinite-dimensional graphs. The basic idea is to enlarge the parameter space by adding an auxiliary parameter (which plays the role of the inverse temperature in the directed polymer formulation) which allows us to fine-tune the effect of anomalously large outliers in the far-tails of the probability distributions of the transmission amplitudes between far-away many-body configurations in the Hilbert space. We first benchmark our approach onto two non-interacting paradigmatic toy models, namely the single-particle Anderson model on the (loop-less) Cayley tree and the Rosenzweig-Porter random matrix ensemble, and then apply it to the study of a class of disordered quantum spin chains in a transverse field. This analysis shows the existence of a broad disorder range in which rare, long-distance resonances, that may form only for a few specific realizations of the disorder and a few specific choice of the random initial state, destabilize the MBL phase, while the genuine MBL transition is shifted to much larger values of the disorder than originally thought.
The effect of rare system-wide resonances in the many-body localization (MBL) transition has recently attracted significant attention. They are expected to play a prominent role in the stability of the MBL phase, prompting the development of new theoretical frameworks to properly account for their statistical weight. We employ a method based on an analogy with mean-field disordered glassy systems to characterize the statistics of transmission amplitudes between distant many-body configurations in Hilbert space, and apply it to the random-field XXZ spin chain. By introducing a Lagrange multiplier, which formally plays the role of an effective temperature controlling the influence of extreme outliers in the heavy-tailed distribution of propagators, we identify three distinct regimes: (i) an ergodic phase with uniform spreading in Hilbert space, (ii) an intermediate regime where delocalization is driven by rare, disorder-dependent long-range resonances, and (iii) a robust MBL phase where such resonances cannot destabilize localization. We derive a finite-size phase diagram in the disorder--interaction plane both in the spin and in the Anderson basis that quantitatively agrees with recent numerical results based on real-space spin-spin correlation functions. We further demonstrate that even infinitesimal interactions can destroy the Anderson insulator at finite disorder, with the critical disorder remaining finite down to small interaction strengths. By visualizing resonant transmission pathways on the Hilbert space graph, we provide a complementary perspective to real-space and spectral probes, revealing how the destabilization of the MBL phase at finite sizes stems from the emergence of resonant paths that become progressively rarer and shorter-ranged deep in the localized phase.
Motivated by recent debates around the many-body localization (MBL) problem, and in particular its stability against systemwide resonances, we investigate long-distance spin-spin correlations across the phase diagram of the random-field XXZ model, with a particular focus on the strong disorder regime. Building on state-of-the-art shift-invert diagonalization techniques, we study the high-energy behavior of transverse and longitudinal correlation functions, computed at the largest possible distance, for a broad range of disorder and interaction strengths. Our results show that while transverse correlations display a fairly stable exponential decay over the entire XXZ phase diagram, longitudinal correlations exhibit markedly different behavior, revealing distinct physical regimes. More precisely, we identify an intermediate disorder region where standard observables show well-converged MBL behavior [Colbois et al., Phys. Rev. Lett. 133, 116502 (2024)] while the distributions of longitudinal correlations reveal unexpected fat tails toward large values. These rare events strongly influence the average decay of longitudinal correlations, which we find to be algebraic in a broad region inside the supposed MBL phase, whereas the typical decay remains mostly exponential. At stronger disorder and weaker interactions, this intermediate regime is replaced by a more conventional exponential decay with short correlation lengths for both typical and average correlators, as expected for standard localization. Our findings shed light on the systemwide instabilities and raise important questions about the impact of such rare but large long-range correlations on the stability of the MBL phase. Finally, we discuss the possible fate of the intermediate region in the context of recent perspectives in the field.
Using the noninteracting Anderson tight-binding model on the Bethe lattice as a toy model for the many-body quantum dynamics, we propose a novel and transparent theoretical explanation of the anomalously slow dynamics that emerges in the bad metal phase preceding the many-body localization (MBL) transition. By mapping the time decorrelation of many-body wave functions onto directed polymers in random media, we show the existence of a glass transition within the extended regime of loopless Cayley trees separating a metalliclike phase at small disorder, where delocalization occurs on an exponential number of paths, from a bad metal-like phase at intermediate disorder, where resonances are formed on rare, specific, disorder dependent site orbitals on very distant generations. The physical interpretation of subdiffusion and nonexponential relaxation emerging from this picture is complementary to the Griffiths one, although both scenarios rely on the presence of heavy-tailed distribution of the escape times. We relate the dynamical evolution in the glassy phase to the depinning transition of directed polymers, which results in macroscopic and abrupt jumps of the preferred delocalizing paths when a parameter like the energy is varied, and produce a singular behavior of the overlap correlation function between eigenstates at different energies. By comparing the quantum dynamics on loopless Cayley trees and random regular graphs, we discuss the effect of loops, showing that in the latter slow dynamics and apparent power laws extend on a very large time window but are eventually cutoff on a timescale that diverges at the MBL transition.
We introduce and study a toy model for anomalous transport and Griffiths effects in one-dimensional (1D) quantum disordered isolated systems near the many-body localization (MBL) transition. The model is constituted by a collection of 1D tight-binding chains with on-site random energies, locally coupled to a weak perturbation, corresponding to a random matrix which belongs to the Gaussian orthogonal ensemble (GOE) and mimics the effect of thermal inclusions due to delocalizing interactions by providing a local broadening of the Poisson spectrum. While in the absence of such a coupling the model is localized as expected for the one-dimensional Anderson model, increasing the coupling with the GOE perturbation we find a transition to a conducting phase driven by the proliferation of quantum avalanches, which does not fit the standard paradigm of the Anderson transition. In particular an intermediate Griffiths region emerges, where exponentially distributed insulating segments coexist with a few rare resonances. Typical correlations decay exponentially fast, while average correlations decay as stretched exponentials and diverge with the length of the chain, indicating that the conducting inclusions have a fractal structure and that the localization length is broadly distributed at the critical point. This behavior is consistent with a Kosterlitz-Thouless-like criticality of the MBL transition. Transport and relaxation are dominated by rare resonances and rare strong insulating regions, and show anomalous behaviors strikingly similar to those observed in recent simulations and experiments in the bad metal delocalized phase preceding MBL. In particular, we find subdiffusive transport and power-law decay of the return probability at large times, with exponents that gradually change as one moves across the intermediate region. Concomitantly, the ac conductivity vanishes near zero frequency with an anomalous power law.
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).