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arXiv:2605.20656

Entanglement Growth from Structured Initial States in Many-Body Localized Systems

Chen Xu, Pengfei Zhang

quant-phcond-mat.dis-nnSubmitted May 20, 2026• Announced May 21, 2026• 0 citations• Revised Oct 12, 2026 (v2)
v1 · May 20, 2026v2 · Oct 12, 2026
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Abstract

Understanding how complex entanglement structures emerge is a central problem in quantum many-body physics. Recent work by Zhang et al. [Phys. Rev. Lett. 137, 020404 (2026)] has considered structured initial states prepared by evolving a product state under a chaotic Hamiltonian for a finite time before quenching to the target Hamiltonian. In this setup, total entanglement entropy growth in many-body localized systems exhibits two distinct regimes, first increasing and then decreasing as the initial entanglement is tuned. In this work, we identify the physical origin of this behavior by analyzing the dynamics of both the Rényi entanglement entropy and the Wehrl-Rényi entropy in the random-field XXZ model, the latter of which characterizes multipartite entanglement. We show that a similar non-monotonic dependence on the initial entanglement also appears in the net growth of the Wehrl-Rényi entropy for product states polarized along the $z$-direction. The first regime is governed by a finite magnetization associated with local integrals of motion, while the second reflects inter-site correlations. In contrast, for product states in the $x/y$-direction, the entanglement growth exhibits a monotonic decay. Our results provide a more fine-grained picture of how distinct initial-state properties shape entanglement dynamics in many-body localized systems.

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Understanding how complex entanglement structures emerge is a central problem in quantum many-body physics. Recent work by Zhang et al. [Phys. Rev. Lett. 137, 020404 (2026)] has considered structured initial states prepared by evolving a product state under a chaotic Hamiltonian for a finite time before quenching to the target Hamiltonian. In this setup, total entanglement entropy growth in many-body localized systems exhibits two distinct regimes, first increasing and then decreasing as the initial entanglement is tuned. In this work, we identify the physical origin of this behavior by analyzing the dynamics of both the Rényi entanglement entropy and the Wehrl-Rényi entropy in the random-field XXZ model, the latter of which characterizes multipartite entanglement. We show that a similar non-monotonic dependence on the initial entanglement also appears in the net growth of the Wehrl-Rényi entropy for product states polarized along the zzz-direction. The first regime is governed by a finite magnetization associated with local integrals of motion, while the second reflects inter-site correlations. In contrast, for product states in the x/yx/yx/y-direction, the entanglement growth exhibits a monotonic decay. Our results provide a more fine-grained picture of how distinct initial-state properties shape entanglement dynamics in many-body localized systems.
The dynamics of entanglement has recently been realized as a useful probe in studying ergodicity and its breakdown in quantum many-body systems. In this paper, we study theoretically the growth of entanglement in quantum many-body systems and propose a method to measure it experimentally. We show that entanglement growth is related to the spreading of local operators in real space. We present a simple toy model for ergodic systems in which linear spreading of operators results in a universal, linear-in-time growth of entanglement for initial product states, in contrast with the logarithmic growth of entanglement in many-body localized (MBL) systems. Furthermore, we show that entanglement growth is directly related to the decay of the Loschmidt echo in a composite system comprised of several copies of the original system, in which connections are controlled by a quantum switch (two-level system). By measuring only the switch's dynamics, the growth of the R\'enyi entropies can be extracted. Our work provides a way of understanding entanglement dynamics in many-body systems and to directly measure its growth in time via a single local measurement.
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.
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We prove rigorous bounds on the growth of $\ensuremath{\alpha}$-R\'enyi entropies ${S}_{\ensuremath{\alpha}}(t)$ (the von Neumann entropy being the special case $\ensuremath{\alpha}=1$) associated with any subsystem AAA of a general lattice quantum many-body system with finite on-site Hilbert space dimension. For completely nonlocal Hamiltonians, we show that the instantaneous growth rates $|{S}_{\ensuremath{\alpha}}^{\ensuremath{'}}(t)|$ (with $\ensuremath{\alpha}\ensuremath{\ne}1$) can be exponentially larger than $|{S}_{1}^{\ensuremath{'}}(t)|$ as a function of the subsystem size ∣A∣|A|∣A∣. For DDD-dimensional systems with geometric locality, we prove bounds on $|{S}_{\ensuremath{\alpha}}^{\ensuremath{'}}(t)|$ that depend on the decay rate of interactions with distance. When $\ensuremath{\alpha}=1$, the bound is ∣A∣|A|∣A∣ independent for all power-law decaying interactions $V(r)\ensuremath{\sim}{r}^{\ensuremath{-}w}$ with w>2D+1w>2D+1w>2D+1. However, for $\ensuremath{\alpha}>1$, the bound is ∣A∣|A|∣A∣ independent only when the interactions are finite-range or decay faster than $V(r)\ensuremath{\sim}{e}^{\ensuremath{-}c\phantom{\rule{0.16em}{0ex}}{r}^{D}}$ for some ccc depending on the local Hilbert space dimension. Using similar arguments, we also prove bounds on kkk-local systems with or without geometric locality. A central theme of this work is that the value of $\ensuremath{\alpha}$ strongly influences the interplay between locality and instantaneous entanglement growth. In other words, the von Neumann entropy and the $\ensuremath{\alpha}$-R\'enyi entropies cannot be regarded as proxies for each other in studies of entanglement dynamics. We compare these bounds with analytic and numerical results on Hamiltonians with varying degrees of locality and find concrete examples that almost saturate the bound for nonlocal dynamics.
Dynamics of many-body localized systems: logarithmic lightcones and log⁡ t\log \, tlogt-law of ααα-Rényi entropies
In the context of the Many-Body-Localization phenomenology we consider arbitrarily large one-dimensional local spin systems, the XXZ model with random magnetic field is a prototypical example. Without assuming the existence of exponentially localized integrals of motion (LIOM), but assuming instead that the system's dynamics gives rise to a Lieb-Robinson bound (L-R) with a logarithmic lightcone, we rigorously evaluate the dynamical generation, starting from a generic product state, of $ α$-Rényi entropies, with $ α$ close to one, obtaining a log⁡ t\log \, tlogt-law, that denotes a slow spread of entanglement. This is in sharp contrast with Anderson localized phases that show no dynamically generated entanglement. To prove this result we apply a general theory recently developed by us in arXiv:2408.00743 that quantitatively relates the L-R bounds of a local Hamiltonian with the dynamical generation of entanglement. Assuming instead the existence of LIOM we provide new independent proofs of the known facts that the L-R bound of the system's dynamics has a logarithmic lightcone and show that the dynamical generation of the von Neumann entropy has for large times a $ \log \, t$-shape. L-R bounds, that quantify the dynamical spreading of local operators, may be easier to measure in experiments in comparison to global quantities such as entanglement.