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arXiv:1404.5216 · Physical Review B

Entanglement spreading in a many-body localized system

Arun Nanduri, HyungWon Kim, David A. Huse

Published Aug 14, 2014• 118 citations
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Abstract

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 $XXZ$ 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.

DOI
10.1103/physrevb.90.064201

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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 (\ensuremathξ)(\ensuremath{\xi})(\ensuremathξ) 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∞)({s}_{\ensuremath{\infty}})(s\ensuremath∞​), which primarily depends on the initial state. The entanglement entropy thus grows with time as \ensuremath∼\ensuremathξ\ifmmode×\else\texttimes\fis\ensuremath∞logt\ensuremath{\sim}\ensuremath{\xi}\ifmmode\times\else\texttimes\fi{}{s}_{\ensuremath{\infty}}logt\ensuremath∼\ensuremathξ\ifmmode×\else\texttimes\fis\ensuremath∞​logt, providing support for the phenomenology of many-body localized systems proposed by Huse and Oganesyan.
We consider fully many-body-localized systems, i.e., isolated quantum systems where all the many-body eigenstates of the Hamiltonian are localized. We define a sense in which such systems are integrable, with localized conserved operators. These localized operators are interacting pseudospins, and the Hamiltonian is such that unitary time evolution produces dephasing but not ``flips'' of these pseudospins. As a result, an initial quantum state of a pseudospin can in principle be recovered via (pseudospin) echo procedures. We discuss how the exponentially decaying interactions between pseudospins lead to logarithmic-in-time spreading of entanglement starting from nonentangled initial states. These systems exhibit multiple different length scales that can be defined from exponential functions of distance; we suggest that some of these decay lengths diverge at the phase transition out of the fully many-body-localized phase while others remain finite.
Recent numerical work by Bardarson, Pollmann, and Moore revealed a slow, logarithmic in time, growth of the entanglement entropy for initial product states in a putative many-body localized phase. We show that this surprising phenomenon results from the dephasing due to exponentially small interaction-induced corrections to the eigenenergies of different states. For weak interactions, we find that the entanglement entropy grows as ξln(Vt/ℏ), where V is the interaction strength, and ξ is the single-particle localization length. The saturated value of the entanglement entropy at long times is determined by the participation ratios of the initial state over the eigenstates of the subsystem. Our work shows that the logarithmic entanglement growth is a universal phenomenon characteristic of the many-body localized phase in any number of spatial dimensions, and reveals a broad hierarchy of dephasing time scales present in such a phase.
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.
We study entanglement dynamics in a diagonal dephasing model in which the strength of interaction decays exponentially with distance---the so-called l-bit model of many-body localization. We calculate the exact expression for entanglement growth with time, finding in addition to a logarithmic growth, a sublogarithmic correction. Provided the l-bit picture correctly describes the many-body localized phase this implies that the entanglement in such systems does not grow (just) as a logarithm of time, as believed so far.
The non-equilibrium dynamics of disordered many-body quantum systems after a global quantum quench unveils important insights about the competition between interactions and disorder, yielding in particular an insightful perspective on many body localization (MBL). Still, the experimentally relevant effect of bond randomness in long-range interacting spin chains on the quantum quench dynamics have so far not been investigated. In this letter, we examine the entanglement entropy growth after a global quench in a quantum spin chain with randomly placed spins and long-range tunable interactions decaying with distance with power ααα. Using a dynamical version of the strong disorder renormalization group (SDRG) we find for α>αcα>α_cα>αc​ that the entanglement entropy grows logarithmically with time and becomes smaller with larger ααα as S(t)=Spln⁡(t)/(2α)S(t) = S_p \ln(t)/(2α)S(t)=Sp​ln(t)/(2α). Here, Sp=2ln⁡2−1S_p= 2 \ln2 -1Sp​=2ln2−1. We use numerical exact diagonalization (ED) simulations to verify our results for system sizes up to $ N\sim 16$ spins, yielding good agreement for sufficiently large α>αc≈1.8α> α_c \approx 1.8α>αc​≈1.8. For α<αcα<α_cα<αc​, we find that the entanglement entropy grows as a power-law with time, S(t)∼tγ(α)S(t)\sim t^{γ(α)}S(t)∼tγ(α) with 0<γ(α)<10<γ(α)<10<γ(α)<1 a decaying function of the interaction exponent ααα.
Many-body localization occurs in isolated quantum systems when Anderson localization persists in the presence of finite interactions. Despite strong evidence for the existence of a many-body localization transition, a reliable extraction of the critical disorder strength is difficult due to a large drift with system size in the studied quantities. In this Letter, we explore two entanglement properties that are promising for the study of the many-body localization transition: the variance of the half-chain entanglement entropy of exact eigenstates and the long time change in entanglement after a local quench from an exact eigenstate. We investigate these quantities in a disordered quantum Ising chain and use them to estimate the critical disorder strength and its energy dependence. In addition, we analyze a spin-glass transition at large disorder strength and provide evidence for it being a separate transition. We, thereby, give numerical support for a recently proposed phase diagram of many-body localization with localization protected quantum order [Huse et al., Phys. Rev. B 88, 014206 (2013).