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arXiv:1911.12370 · Physical review. B./Physical review. B

Fock-space correlations and the origins of many-body localization

Sthitadhi Roy, David E. Logan

Published Apr 23, 2020• 69 citations
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Abstract

The authors approach the problem of many-body localization on the Fock space, with nodes and links representing many-body states and Hamiltonian matrix-elements. The problem maps exactly onto a tight-binding model on a complex high-dimensional graph with nontrivial correlated disorder. These correlations are shown to lie at the heart of many-body localization. The central result from this theory is that the correlations in the Fock-space disorder must be maximal for a many-body localized phase to be stable.

DOI
10.1103/physrevb.101.134202

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The authors approach the problem of many-body localization on the Fock space, with nodes and links representing many-body states and Hamiltonian matrix-elements. The problem maps exactly onto a tight-binding model on a complex high-dimensional graph with nontrivial correlated disorder. These correlations are shown to lie at the heart of many-body localization. The central result from this theory is that the correlations in the Fock-space disorder must be maximal for a many-body localized phase to be stable.
Models of many-body localization (MBL) can be represented as tight-binding models in the many-body Hilbert space (Fock space). We explore the role of correlations between matrix elements of the effective Fock-space Hamiltonians in the scaling of MBL critical disorder Wc(n){W}_{c}(n)Wc​(n) with the size nnn of the system. For this purpose, we consider five models, which all have the same distributions of diagonal (energy) and off-diagonal (``hopping'') Fock-space matrix elements but different Fock-space correlations. These include quantum-dot (QD) and one-dimensional (1D) MBL models, their modifications (uQD and u1D models) with removed correlations of off-diagonal matrix elements, as well a quantum random energy model (QREM) with no correlations at all. Our numerical results are in full consistency with analytical arguments predicting ${n}^{3/4}{(lnn)}^{\ensuremath{-}1/4}\ensuremath{\lesssim}{W}_{c}\ensuremath{\lesssim}nlnn$ for the scaling of Wc(n){W}_{c}(n)Wc​(n) in the QD model (we find ${W}_{c}\ensuremath{\sim}n$ numerically), ${W}_{c}(n)\ensuremath{\sim}\text{const}$ for the 1D model, ${W}_{c}\ensuremath{\sim}nlnn$ for the uQD and u1D models without off-diagonal correlations, and ${W}_{c}\ensuremath{\sim}{n}^{1/2}lnn$ for QREM. The key difference between the QD and 1D models is in the structure of correlations of many-body energies. Removing off-diagonal Fock-space correlations makes both these models ``maximally chaotic''. Our findings demonstrate that the scaling of Wc(n){W}_{c}(n)Wc​(n) for MBL transitions is governed by a combined effect of Fock-space correlations of diagonal and off-diagonal matrix elements.
This article reviews recent progress in understanding the physics of many-body localisation (MBL) in disordered and interacting quantum many-body systems, from the perspective of ergodicity breaking on the associated Fock space. This approach to MBL is underpinned by mapping the dynamics of the many-body system onto that of a fictitious single particle on the high-dimensional, correlated and disordered Fock-space graph; yet, as we elaborate, the problem is fundamentally different from that of conventional Anderson localisation on high-dimensional or hierarchical graphs. We discuss in detail the nature of eigenstate correlations on the Fock space, both static and dynamic, and in the ergodic and many-body localised phases as well as in the vicinity of the MBL transition. The latter in turn sheds light on the nature of the transition, and motivates a scaling theory for it in terms of Fock-space based quantities. We also illustrate how these quantities can be concretely connected to real-space observables. An overview is given of several analytical and numerical techniques which have proven important in developing a comprehensive picture. Finally, we comment on some open questions in the field of MBL where the Fock-space approach is likely to prove insightful.
In the presence of strong enough disorder one-dimensional systems of interacting spinless fermions at nonzero filling factor are known to be in a many-body localized phase. When represented in `Fock space,' the Hamiltonian of such a system looks like that of a single `particle' hopping on a Fock lattice in the presence of a random disordered potential. The coordination number of the Fock lattice increases linearly with the system size LLL in one dimension. Thus in the thermodynamic limit $L\ensuremath{\rightarrow}\ensuremath{\infty}$, the disordered interacting problem in one dimension maps on to an Anderson model with infinite coordination number. Despite this, this system displays localization which appears counterintuitive. A close observation of the on-site disorder potentials on the Fock lattice reveals a large degree of correlation among them as they are derived from an exponentially smaller number of on-site disorder potentials in real space. This indicates that the correlations between the on-site disorder potentials on a Fock lattice has a strong effect on the localization properties of the corresponding many-body system. This intuition is also consistent with studies of quantum random energy model where the typical mid-spectrum states are ergodic and the on-site potentials in Fock space are completely uncorrelated. In this work we perform a systematic quantitative exploration of the nature of correlations of the Fock space potential required for localization. We study different functional variations of the disorder correlation in Fock lattice by analyzing the eigenspectrum obtained through exact diagonalization. Without changing the typical strength of the on-site disorder potential in Fock lattice we show that changing the correlation strength can induce thermalization or localization in systems. From among the various forms of correlations we study, we find that only the linear variation of correlations with Hamming distance in Fock space is able to drive a thermal-MBL phase transition where the transition is driven by the correlation strength. Systems with the other forms of correlations we study are found to be ergodic.
A canonical model for many-body localization (MBL) is studied, of interacting spinless fermions on a lattice with uncorrelated quenched site disorder. The model maps onto a tight-binding model on a `Fock-space (FS) lattice' of many-body states, with an extensive local connectivity. We seek to understand some aspects of MBL from this perspective, via local propagators for the FS lattice and their self-energies (SE's), focusing on the SE probability distributions, over disorder and FS sites. A probabilistic mean-field theory (MFT) is first developed, centered on self-consistent determination of the geometric mean of the distribution. Despite its simplicity this captures some key features of the problem, including recovery of an MBL transition, and predictions for the forms of the SE distributions. The problem is then studied numerically in 1d1d1d by exact diagonalization, free from MFT assumptions. The geometric mean indeed appears to act as a suitable order parameter for the transition. Throughout the MBL phase the appropriate SE distribution is confirmed to have a universal form, with long-tailed L\'evy behavior as predicted by MFT. In the delocalized phase for weak disorder, SE distributions are clearly log-normal, while on approaching the transition they acquire an intermediate L\'evy-tail regime, indicative of the incipient MBL phase.
Many-body localization in interacting quantum systems can be cast as a disordered hopping problem on the underlying Fock-space graph. A crucial feature of the effective Fock-space disorder is that the Fock-space site energies are strongly correlated-maximally so for sites separated by a finite distance on the graph. Motivated by this, and to understand the effect of such correlations more fundamentally, we study Anderson localization on Cayley trees and random regular graphs, with maximally correlated disorder. Since such correlations suppress short distance fluctuations in the disorder potential, one might naively suppose they disfavor localization. We find however that there exists an Anderson transition, and indeed that localization is more robust in the sense that the critical disorder scales with graph connectivity K as sqrt[K], in marked contrast to KlnK in the uncorrelated case. This scaling is argued to be intimately connected to the stability of many-body localization. Our analysis centers on an exact recursive formulation for the local propagators as well as a self-consistent mean-field theory; with results corroborated using exact diagonalization.
Drawing motivation from the fact that eigenstates in many-body localized and ergodic phases behave differently with regard to their support on the Fock space, the authors study a classical percolation problem in Fock space that captures certain aspects of the many-body localization transition. In particular, it has two classical phases that provide simplified pictures of the extended and many-body localized states. At weak disorder, the entire Fock space is contained in a single cluster, whereas for strong disorder, the typical cluster size scales with the Fock-space dimension in a subvolume law fashion.