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arXiv:2504.02815 · Physical review. B./Physical review. B

Logarithmic entanglement light cone from eigenstate correlations in the many-body localized phase

Ratul Thakur, Bikram Pain, Sthitadhi Roy

Published May 20, 2025• 2 citations
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Abstract

We investigate the operator entanglement of the time-evolution operator through the framework of eigenstate correlations. Focusing on strongly disordered quantum many-body systems in the many-body localized (MBL) regime, we analyze the operator entanglement across various spatiotemporal cuts, revealing the logarithmic lightcone of entanglement spreading. We demonstrate that this logarithmic lightcone arises directly from a hierarchy of energyscales and lengthscales encoded in eigenstate correlations. By characterizing the statistics of these hierarchical scales, we develop a microscopic theory for the spatiotemporal structure of entanglement spreading in MBL systems---without invoking phenomenological constructs such as $\ensuremath{\ell}$-bits. This approach reveals the fundamental connection between eigenstate correlations and the emergent entanglement structure in MBL systems.

DOI
10.1103/physrevb.111.174206

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We investigate the operator entanglement of the time-evolution operator through the framework of eigenstate correlations. Focusing on strongly disordered quantum many-body systems in the many-body localized (MBL) regime, we analyze the operator entanglement across various spatiotemporal cuts, revealing the logarithmic lightcone of entanglement spreading. We demonstrate that this logarithmic lightcone arises directly from a hierarchy of energyscales and lengthscales encoded in eigenstate correlations. By characterizing the statistics of these hierarchical scales, we develop a microscopic theory for the spatiotemporal structure of entanglement spreading in MBL systems---without invoking phenomenological constructs such as \ensuremathℓ\ensuremath{\ell}\ensuremathℓ-bits. This approach reveals the fundamental connection between eigenstate correlations and the emergent entanglement structure in MBL systems.
The many-body localized phase of quantum systems is an unusual dynamical phase wherein the system fails to thermalize and yet, entanglement grows unboundedly albeit very slowly in time. We present a microscopic theory of this ultraslow growth of entanglement in terms of dynamical eigenstate correlations of strongly disordered, interacting quantum systems in the many-body localized regime. These correlations involve sets of four or more eigenstates and hence, go beyond correlations involving pairs of eigenstates which are usually studied in the context of eigenstate thermalization or lack thereof. We consider the minimal case, namely, the second R\'enyi entropy of entanglement, of an initial product state as well as that of the time-evolution operator, wherein the correlations involve quartets of four eigenstates. We identify that the dynamics of the entanglement entropy is dominated by the spectral correlations within certain special quartets of eigenstates. We uncover the spatial structure of these special quartets and the ensuing statistics of the spectral correlations amongst the eigenstates therein, which reveals a hierarchy of timescales or equivalently, energy scales. We show that the hierarchy of these timescales along with their nontrivial distributions conspire to produce the logarithmic in time growth of entanglement, characteristic of the many-body localized regime. The underlying spatial structures in the set of special quartets also provides a microscopic understanding of the spacetime picture of the entanglement growth. The theory therefore provides a much richer perspective on entanglement growth in strongly disordered systems compared to the commonly employed phenomenological approach based on the $\ensuremath{\ell}$-bit picture.
Motivated by the findings of logarithmic spreading of entanglement in a many-body localized system, we identify and untangle two factors contributing to the spreading of entanglement in the fully many-body localized phase, where all many-body eigenstates are localized. Performing full diagonalizations of an XXZXXZXXZ spin model with random longitudinal fields, we demonstrate a linear dependence of the spreading rate on the decay length $(\ensuremath{\xi})$ of the effective interaction between localized pseudospins (l-bits), which depends on the disorder strength, and on the final value of entanglement per spin $({s}_{\ensuremath{\infty}})$, which primarily depends on the initial state. The entanglement entropy thus grows with time as $\ensuremath{\sim}\ensuremath{\xi}\ifmmode\times\else\texttimes\fi{}{s}_{\ensuremath{\infty}}logt$, providing support for the phenomenology of many-body localized systems proposed by Huse and Oganesyan.
In many-body localized systems, propagation of information forms a light cone that grows logarithmically with time. However, local changes in energy or other conserved quantities are expected to spread only within a finite distance. Is it possible to detect the logarithmic light cone using local observables, starting from thermal states or eigenstates? Here we show that this is indeed the case, but only in an out-of-time-ordered way. In particular, we numerically calculate various correlators in a prototypical model of many-body localization. While correlators of local operators A(t=0)A(t=0)A(t=0) and B(t>0)B(t>0)B(t>0) ordered in time sequence -- e.g., A(t=0)B(t>0)A(t=0)B(t>0)A(t=0)B(t>0) -- are not sensitive to the unbounded information propagation, those with out-of-time order -- e.g., A(t=0)B(t>0)A(t=0)B(t>0)A(t=0)B(t>0)A(t=0)B(t>0)A(t=0)B(t>0)A(t=0)B(t>0) -- can detect the logarithmic light cone. We demonstrate the connection between out-of-time-ordered correlators and the Lieb-Robinson bound in many-body localized systems, and explain how to saturate the latter by measuring the involved commutator on specially designed initial states. Moreover, out-of-time-ordered correlators also allow us to study the temperature dependence of the light cones and we observe an increase of propagation velocity with increasing temperature.
Many-body localization (MBL) hinders the thermalization of quantum many-body systems in the presence of strong disorder. In this Letter, we study the MBL regime in bond-disordered spin-1/2 XXZ spin chain, finding the multimodal distribution of entanglement entropy in eigenstates, sub-Poissonian level statistics, and revealing a relation between operators and initial states required for examining the breakdown of thermalization in the time evolution of the system. We employ a real space renormalization group scheme to identify these phenomenological features of the MBL regime that extend beyond the standard picture of local integrals of motion relevant for systems with disorder coupled to on-site operators. Our results pave the way for experimental probing of MBL in bond-disordered spin chains.
A prime characterization of many-body localized (MBL) systems is the entanglement of their eigenstates; in contrast to the typical ergodic phase whose eigenstates are volume law, MBL eigenstates obey an area law. In this paper, we show that a spin-disordered Hubbard model has both a large number of area-law eigenstates as well as a large number of eigenstates whose entanglement scales logarithmically with system size (log-law). This model provides a microscopic Hamiltonian which is neither ergodic nor MBL. We establish these results through a combination of analytic arguments based on the eta-pairing operators combined with a numerical analysis of eigenstates. In addition, we describe and simulate a dynamic time evolution approach starting from product states through which one can separately probe the area-law and log-law eigenstates in this system.
Effective light cones, characterized by Lieb-Robinson bounds, emerge in nonrelativistic local quantum systems. Here, we present several analytical results derived from logarithmic light cones (LLCs). Possible origins of LLCs include the one-dimensional (1D) disordered XXZ model and a phenomenological model of many-body localization (MBL). In the LLC regime, we prove that, for arbitrary spatial dimensions and any initial pure state, entanglement growth is upper bounded by logarithmic time with an additional subleading double-logarithmic correction---arising from a real asymptotic solution of the Lambert W function---valid up to the asymptotic time limit. In the context of the 1D disordered XXZ model, this result resolves the ambiguity in distinguishing between logarithmic and power-law fits of entanglement growth in numerical studies; we also propose a falsifiable phenomenological functional form for the entanglement growth that agrees with existing numerical results. We show that information scrambling is logarithmically slow in the LLC regime. Furthermore, we demonstrate that the LLC supports long-lived quantum memories---quantum codes with macroscopic code distance and lifetimes that scale exponentially with system size---under unitary time evolution. Our analytical results provide benchmarks for future numerical studies of the MBL regime at large timescales.