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arXiv:2204.13804 · Physical review. B./Physical review. B

Critical localization with van der Waals interactions

Rahul Nandkishore

Published Aug 24, 2022• 0 citations
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Abstract

I discuss the quantum dynamics of strongly disordered quantum systems with critically long range interactions, decaying as $1/{r}^{2d}$ in $d$ spatial dimensions. I argue that, contrary to expectations, localization in such systems is stable at low orders in perturbation theory, giving rise to an unusual ``critically many-body localized (MBL) regime.'' I discuss the phenomenology of this critical MBL regime, which includes distinctive signatures in entanglement, charge statistics, noise, and transport. Experimentally, such a critically localized regime can be realized in three-dimensional systems with van der Waals interactions, such as Rydberg atoms, and in one-dimensional systems with $1/{r}^{2}$ interactions, such as trapped ions. I estimate timescales on which high-order perturbative and nonperturbative (avalanche) phenomena may destabilize this critically MBL regime and conclude that the avalanche sets the limiting timescale, in the limit of strong disorder or weak interactions.

DOI
10.1103/physrevb.106.l060306

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I discuss the quantum dynamics of strongly disordered quantum systems with critically long range interactions, decaying as 1/r2d1/{r}^{2d}1/r2d in ddd spatial dimensions. I argue that, contrary to expectations, localization in such systems is stable at low orders in perturbation theory, giving rise to an unusual ``critically many-body localized (MBL) regime.'' I discuss the phenomenology of this critical MBL regime, which includes distinctive signatures in entanglement, charge statistics, noise, and transport. Experimentally, such a critically localized regime can be realized in three-dimensional systems with van der Waals interactions, such as Rydberg atoms, and in one-dimensional systems with 1/r21/{r}^{2}1/r2 interactions, such as trapped ions. I estimate timescales on which high-order perturbative and nonperturbative (avalanche) phenomena may destabilize this critically MBL regime and conclude that the avalanche sets the limiting timescale, in the limit of strong disorder or weak interactions.
We investigate the stability of the many-body localized phase against quantum avalanche instabilities in a one-dimensional Heisenberg spin chain with long-range power-law interactions (V∝r−αV\propto r^{-α}V∝r−α). By combining exact diagonalization of static properties with Lindblad master equation simulations of open-system dynamics, we systematically map the interplay between interaction range and disorder strength. Our finite-size scaling analysis of entanglement entropy identifies a critical interaction exponent αc≈2α_c \approx 2αc​≈2, which separates a fragile regime, characterized by an exponentially diverging critical disorder, from a robust short-range regime. To rigorously test the system's resistance to avalanches, we couple the boundary to an infinite-temperature bath and track the propagation of the thermalization front into the localized bulk. We find that the characteristic thermalization time follows a unified scaling law, Trth∼exp⁡[κ(α)LW]T_{r_{\text{th}}} \sim \exp[κ(α) LW]Trth​​∼exp[κ(α)LW] (herein, LLL is the system size, and WWW is the disorder intensity), which diverges exponentially with the product of system size and disorder strength. This suppression enables the derivation of a quantitative stability criterion, Wstab(α)W_{\text{stab}}(α)Wstab​(α), representing the minimum critical disorder strength required to maintain avalanche stability. Our results confirm that the MBL phase remains asymptotically stable in the thermodynamic limit when disorder exceeds an interaction-dependent threshold, bridging theoretical debates on long-range MBL and providing a roadmap for observing these dynamics in experimental platforms such as Rydberg atom arrays.
We investigate the many-body localization (MBL) transition in the quantum Ising model with long-range interactions. Unlike spin chains with short-range interactions, where the MBL transition point remains elusive due to strong finite-size effects and local fluctuations, long-range interactions suppress such fluctuations and enable clearer signatures of critical behavior. Using exact results from a related Bethe lattice localization problem, we estimate the MBL threshold within logarithmic accuracy and find consistency with exact diagonalization. Although the critical disorder diverges in the thermodynamic limit, our results demonstrate that the critical regime can still be probed and highlight the relevance of this model to systems with dipole-dipole, elastic, or indirect exchange interactions.
We discuss whether localization in the two-dimensional continuum can be stable in the presence of short-range interactions. We conclude that, for an impurity model of disorder, if the system is prepared below a critical temperature T<TcT<{T}_{c}T<Tc​, then perturbation theory about the localized phase converges almost everywhere. As a result, the system is at least asymptotically localized and perhaps even truly many-body localized, depending on how certain rare regions behave. Meanwhile, for T>TcT>{T}_{c}T>Tc​, perturbation theory fails to converge, which we interpret as interaction-mediated delocalization. We calculate the boundary of the region of perturbative stability of localization in the interaction-strength-temperature plane. We also discuss the behavior in a speckle disorder (relevant for cold-atom experiments) and conclude that perturbation theory about the noninteracting phase diverges for arbitrarily weak interactions with speckle disorder, suggesting that many-body localization in the two-dimensional continuum cannot survive away from the impurity limit.
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).
Abstract Statistical mechanics provides a framework for describing the physics of large, complex many-body systems using only a few macroscopic parameters to determine the state of the system. For isolated quantum many-body systems, such a description is achieved via the eigenstate thermalization hypothesis (ETH), which links thermalization, ergodicity and quantum chaotic behavior. However, tendency towards thermalization is not observed at finite system sizes and evolution times in a robust many-body localization (MBL) regime found numerically and experimentally in the dynamics of interacting many-body systems at strong disorder. Although the phenomenology of the MBL regime is well-established, the central question remains unanswered: under what conditions does the MBL regime give rise to an MBL phase , in which the thermalization does not occur even in the asymptotic limit of infinite system size and evolution time? This review focuses on recent numerical investigations aiming to clarify the status of the MBL phase, and it establishes the critical open questions about the dynamics of disordered many-body systems. The last decades of research have brought an unprecedented new variety of tools and indicators to study the breakdown of ergodicity, ranging from spectral and wave function measures, matrix elements of observables, through quantities probing unitary quantum dynamics, to transport and quantum information measures. We give a comprehensive overview of these approaches and attempt to provide a unified understanding of their main features. We emphasize general trends towards ergodicity with increasing length and time scales, which exclude naive single-parameter scaling hypothesis, necessitate the use of more refined scaling procedures, and prevent unambiguous extrapolations of numerical results to the asymptotic limit. Providing a concise description of numerical methods for studying ETH and MBL, we explore various approaches to tackle the question of the MBL phase. Persistent finite size drifts towards ergodicity consistently emerge in quantities derived from eigenvalues and eigenvectors of disordered many-body systems. The drifts are related to continuous inching towards ergodicity and non-vanishing transport observed in the dynamics of many-body systems, even at strong disorder. These phenomena impede the understanding of microscopic processes at the ETH-MBL crossover. Nevertheless, the abrupt slowdown of dynamics with increasing disorder strength provides premises suggesting the proximity of the MBL phase. This review concludes that the questions about thermalization and its failure in disordered many-body systems remain a captivating area open for further explorations.
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.