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arXiv:2305.10574 · Physical review. B./Physical review. B

Breaking the chains: Extreme value statistics and localization in random spin chains

Jeanne Colbois, Nicolas Laflorencie

Published Oct 24, 2023• 10 citations
arXivHTMLPDF

Abstract

Despite a very good understanding of single-particle Anderson localization in one-dimensional (1D) disordered systems, many-body effects are still full of surprises; a famous example is the interaction-driven many-body localization (MBL) problem, about which much has been written, and perhaps the best is yet to come. Interestingly enough the noninteracting limit provides a natural playground to study nontrivial multiparticle physics, offering the possibility to test some general mechanisms with very large-scale exact diagonalization simulations. In this paper, we first revisit the 1D many-body Anderson insulator through the lens of extreme value theory, focusing on the extreme polarizations of the equivalent spin chain model in a random magnetic field. A many-body-induced chain breaking mechanism is explored numerically, and compared to an analytically solvable toy model. A unified description, from weak to large disorder strengths $W$ emerges, where the disorder-dependent average localization length $\ensuremath{\xi}(W)$ governs the extreme events leading to chain breaks. In particular, tails of the local magnetization distributions are controlled by $\ensuremath{\xi}(W)$. Remarkably, we also obtain a quantitative understanding of the full distribution of the extreme polarizations, which is given by a Fr\'echet-type law. In a second part, we explore finite interaction physics and the MBL question. For the available system sizes, we numerically quantify the difference in the extreme value distributions between the interacting problem and the noninteracting Anderson case. Strikingly, we observe a sharp ``extreme-statistics transition'' as $W$ changes, which may coincide with the MBL transition.

DOI
10.1103/physrevb.108.144206

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Despite a very good understanding of single-particle Anderson localization in one-dimensional (1D) disordered systems, many-body effects are still full of surprises; a famous example is the interaction-driven many-body localization (MBL) problem, about which much has been written, and perhaps the best is yet to come. Interestingly enough the noninteracting limit provides a natural playground to study nontrivial multiparticle physics, offering the possibility to test some general mechanisms with very large-scale exact diagonalization simulations. In this paper, we first revisit the 1D many-body Anderson insulator through the lens of extreme value theory, focusing on the extreme polarizations of the equivalent spin chain model in a random magnetic field. A many-body-induced chain breaking mechanism is explored numerically, and compared to an analytically solvable toy model. A unified description, from weak to large disorder strengths WWW emerges, where the disorder-dependent average localization length \ensuremathξ(W)\ensuremath{\xi}(W)\ensuremathξ(W) governs the extreme events leading to chain breaks. In particular, tails of the local magnetization distributions are controlled by \ensuremathξ(W)\ensuremath{\xi}(W)\ensuremathξ(W). Remarkably, we also obtain a quantitative understanding of the full distribution of the extreme polarizations, which is given by a Fr'echet-type law. In a second part, we explore finite interaction physics and the MBL question. For the available system sizes, we numerically quantify the difference in the extreme value distributions between the interacting problem and the noninteracting Anderson case. Strikingly, we observe a sharp ``extreme-statistics transition'' as WWW changes, which may coincide with the MBL transition.
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.
Recent studies of delocalization-localization transitions in disordered quantum chains have highlighted the role of rare, chain-breaking events that favor localization, in particular for high-energy eigenstates related to many-body localization. In this context, we revisit the random-field XXZ spin-1/2 chain at zero temperature with ferromagnetic interactions, equivalent to interacting fermions or hard-core bosons in a random potential with attractive interactions. We argue that localization in this model can be characterized by chain-breaking events, which are probed by the extreme values of simple local observables, such as the on-site density or the local magnetization, that are readily accessible in both experiments and numerical simulations. Adopting a bosonic language, we study the disorder-induced Berezinskii-Kosterlitz-Thouless (BKT) quantum phase transition from superfluid (SF) to Bose glass (BG), and focus on the strong disorder regime where localization is driven by weak links. Based on high-precision density matrix renormalization group simulations, we numerically show that extreme local densities accurately capture the BKT transition, even for relatively short chains ranging from a few dozen to a hundred sites. We also discuss the SF-BG transition in the weak disorder regime, where finite-size effects pose greater challenges. Overall, our work seeks to establish a solid foundation for using extreme statistics of local observables, such as density, to probe delocalization-localization transitions in disordered quantum chains, both in the ground state and at high energy.
Despite enormous efforts devoted to the study of the many-body localization (MBL) phenomenon, the nature of the high-energy behavior of the Heisenberg spin chain in a strong random magnetic field is lacking consensus. Here, we take a step back by exploring the weak interaction limit starting from the Anderson localized (AL) insulator. Through shift-invert diagonalization, we find that, below a certain disorder threshold h^{*}, weak interactions necessarily lead to an ergodic instability, whereas at strong disorder the AL insulator directly turns into MBL, in agreement with a simple interpretation of the avalanche theory for restoration of ergodicity. We further map the phase diagram for the generic XXZ model in the disorder h-interaction Δ plane. Taking advantage of the total magnetization conservation, our results unveil the remarkable behavior of the spin-spin correlation functions: in the regime indicated as MBL by standard observables, their exponential decay undergoes an inversion of orientation ξ_{z}>ξ_{x}. We find that the longitudinal length ξ_{z} is a key quantity for capturing ergodic instabilities, as it increases with system size near the thermal phase, in sharp contrast to its transverse counterpart.
Ergodic quantum many-body systems satisfy the eigenstate thermalization hypothesis (ETH). However, strong disorder can destroy ergodicity through many-body localization (MBL) – at least in one dimensional systems – leading to a clear signal of the MBL transition in the probability distributions of energy eigenstate expectation values of local operators. For a paradigmatic model of MBL, namely the random-field Heisenberg spin chain, we consider the full probability distribution of eigenstate correlation functions across the entire phase diagram. We find gaussian distributions at weak disorder, as predicted by pure ETH. At intermediate disorder – in the thermal phase – we find further evidence for anomalous thermalization in the form of heavy tails of the distributions. In the MBL phase, we observe peculiar features of the correlator distributions: a strong asymmetry in S_i^z S_{i+r}^z S i z S i + r z correlators skewed towards negative values; and a multimodal distribution for spin-flip correlators. A quantitative quasi-degenerate perturbation theory calculation of these correlators yields a surprising agreement of the full distribution with the exact results, revealing, in particular, the origin of the multiple peaks in the spin-flip correlator distribution as arising from the resonant and off-resonant admixture of spin configurations. The distribution of the S_i^zS_{i+r}^z S i z S i + r z correlator exhibits striking differences between the MBL and Anderson insulator cases.
We theoretically investigate the many-body localization phase transition in a one-dimensional Ising spin chain with random long-range spin-spin interactions, ${V}_{ij}\ensuremath{\propto}{\left|i\ensuremath{-}j\right|}^{\ensuremath{-}\ensuremath{\alpha}}$, where the exponent of the interaction range $\ensuremath{\alpha}$ can be tuned from zero to infinitely large. By using exact diagonalization, we calculate the half-chain entanglement entropy and the energy spectral statistics and use them to characterize the phase transition towards the many-body localization phase at infinite temperature and at sufficiently large disorder strength. We perform finite-size scaling to extract the critical disorder strength and the critical exponent of the divergent localization length. With increasing $\ensuremath{\alpha}$, the critical exponent experiences a sharp increase at about ${\ensuremath{\alpha}}_{c}\ensuremath{\simeq}1.2$ and then gradually decreases to a value found earlier in a disordered short-ranged interacting spin chain. For $\ensuremath{\alpha}<{\ensuremath{\alpha}}_{c}$, we find that the system is mostly localized and the increase in the disorder strength may drive a transition between two many-body localized phases. In contrast, for $\ensuremath{\alpha}>{\ensuremath{\alpha}}_{c}$, the transition is from a thermalized phase to the many-body localization phase. Our predictions could be experimentally tested with an ion-trap quantum emulator with programmable random long-range interactions, or with randomly distributed Rydberg atoms or polar molecules in lattices.
We study the effects of a random magnetic field on a one-dimensional (1D) spin-1 chain with correlated nearest-neighbor XY interaction. We show that this spin model can be exactly mapped onto the 1D disordered tight-binding model of Z3{\mathbb{Z}}_{3}Z3​ Fock parafermions (FPFs), exotic anyonic quasiparticles that generalize usual spinless fermions. Thus, we have a peculiar case of a disordered Hamiltonian that despite being bilinear in the creation and annihilation operators, exhibits a many-body localization (MBL) transition owing to the nontrivial statistics of FPFs. This is in sharp contrast to conventional bosonic and fermionic quadratic disordered Hamiltonians that show single-particle (Anderson) localization. We perform finite-size exact diagonalization calculations of level-spacing statistics, fractal dimensions, and entanglement entropy, and provide convincing evidence for the MBL transition at finite disorder strength.