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arXiv:2010.07969

Comment on "Entanglement growth in diffusive systems"

Tibor Rakovszky, Frank Pollmann, Curt W. von Keyserlingk

Published Oct 15, 2020• 5 citations
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Abstract

In a recent paper (Commun. Phys. 3, 100) Znidaric studies the growth of higher Renyi entropies in diffusive systems and claims that they generically grow ballistically in time, except for spin-1/2 models in d=1 dimension. Here, we point out that the necessary conditions for sub-ballistic growth of Renyi entropies are in fact much more general, and apply to a large class of systems, including experimentally relevant ones in arbitrary dimension and with larger local Hilbert spaces.

DOI
10.48550/arxiv.2010.07969

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Recent works 1 , 2 argued that Rényi entropies S α with indices α > 1 exhibit a sub-ballistic, \(\propto \sqrt{t}\) , growth in systems with diffusive transport. A subsequent work by Žnidarič 3 claims that such sub-ballistic growth occurs only in certain cases (in particular, d = 1 dimensional systems with q = 2 states per site) and is generically replaced by ballistic growth in the absence of further fine-tuning. Below, we try to make precise the conditions needed for diffusive (rather than ballistic) Rényi growth and argue that they apply to a much wider class of systems than what is suggested by Žnidarič 3 . In particular, the examples presented by Žnidarič that avoid diffusive growth do so because of the presence of additional non-conserved degrees of freedom, not merely their larger local Hilbert space.

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Reply to ‘Entanglement growth in diffusive systems with large spin’

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The question discussed 1 , 2 is about the growth of the second Rényi entropy S 2 at long times in systems with a diffusive degree of freedom (DOF). One in particular wants to distinguish between the diffusive \({S}_{2} \sim \sqrt{t}\) , found for any diffusive system with the local Hilbert space dimension q = 2 and spatial dimension 3 d = 1, and ballistic S 2 ~ t growth that is generic 2 for q > 2. Contested 1 is the case of single diffusive DOF like charge, density, or spin in systems with q > 2.

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In a recent paper (Commun. Phys. 3, 100) Znidaric studies the growth of higher Renyi entropies in diffusive systems and claims that they generically grow ballistically in time, except for spin-1/2 models in d=1 dimension. Here, we point out that the necessary conditions for sub-ballistic growth of Renyi entropies are in fact much more general, and apply to a large class of systems, including experimentally relevant ones in arbitrary dimension and with larger local Hilbert spaces.
Recent works 1 , 2 argued that Rényi entropies S α with indices α > 1 exhibit a sub-ballistic, ∝t\propto \sqrt{t}∝t​ , growth in systems with diffusive transport. A subsequent work by Žnidarič 3 claims that such sub-ballistic growth occurs only in certain cases (in particular, d = 1 dimensional systems with q = 2 states per site) and is generically replaced by ballistic growth in the absence of further fine-tuning. Below, we try to make precise the conditions needed for diffusive (rather than ballistic) Rényi growth and argue that they apply to a much wider class of systems than what is suggested by Žnidarič 3 . In particular, the examples presented by Žnidarič that avoid diffusive growth do so because of the presence of additional non-conserved degrees of freedom, not merely their larger local Hilbert space.
The question discussed 1 , 2 is about the growth of the second Rényi entropy S 2 at long times in systems with a diffusive degree of freedom (DOF). One in particular wants to distinguish between the diffusive S2∼t{S}_{2} \sim \sqrt{t}S2​∼t​ , found for any diffusive system with the local Hilbert space dimension q = 2 and spatial dimension 3 d = 1, and ballistic S 2 ~ t growth that is generic 2 for q > 2. Contested 1 is the case of single diffusive DOF like charge, density, or spin in systems with q > 2.
We investigate the dynamics of quantum entanglement after a global quench and uncover a qualitative difference between the behavior of the von Neumann entropy and higher Rényi entropies. We argue that the latter generically grow sub-ballistically, as ∝sqrt[t], in systems with diffusive transport. We provide strong evidence for this in both a U(1) symmetric random circuit model and in a paradigmatic nonintegrable spin chain, where energy is the sole conserved quantity. We interpret our results as a consequence of local quantum fluctuations in conserved densities, whose behavior is controlled by diffusion, and use the random circuit model to derive an effective description. We also discuss the late-time behavior of the second Rényi entropy and show that it exhibits hydrodynamic tails with three distinct power laws occurring for different classes of initial states.
Abstract Entanglement helps in understanding diverse phenomena, going from quantifying complexity to classifying phases of matter. Here we study the influence of conservation laws on entanglement growth. Focusing on systems with U(1) symmetry, i.e., conservation of charge or magnetization, that exhibits diffusive dynamics, we theoretically predict the growth of entanglement, as quantified by the Rényi entropy, in lattice systems in any spatial dimension d and for any local Hilbert space dimension q (qudits). We find that the growth depends both on d and q, and is in generic case first linear in time, similarly as for systems without any conservation laws. Exception to this rule are chains of 2-level systems where the dependence is a square-root of time at all times. Predictions are numerically verified by simulations of diffusive Clifford circuits with upto ~ 105 qubits. Such efficiently simulable circuits should be a useful tool for other many-body problems.
Abstract My previous work [arXiv: 1902.00977 ] studied the dynamics of Rényi entanglement entropy R α in local quantum circuits with charge conservation. Initializing the system in a random product state, it was proved that R α with Rényi index α > 1 grows no faster than ‘diffusively’ (up to a sublogarithmic correction) if charge transport is not faster than diffusive. The proof was given only for qubit or spin-1/2 systems. In this note, I extend the proof to qudit systems, i.e., spin systems with local dimension d ≥ 2 .
The notion of operator growth in quantum systems furnishes a bridge between transport and the generation of entanglement between different parts of the system under quantum dynamics. We define a measure of operator growth in terms of subsystem operator Rényi entropy, which provides a state-independent measure of operator growth, unlike entanglement entropies, and the usual measures of operator growth like out-of-time-order correlators. We show that the subsystem operator Rényi entropy encodes both spatial and temporal information, and thus can directly connect to transport for a local operator related to a conserved quantity. We construct a unified Schwinger-Keldysh (SK) field theory formalism for the time evolution of operator Rényi entropy and entanglement entropies of initial pure states. We use the SK field theory to obtain the operator Rényi and state entanglement entropies in terms of infinite-temperature and vacuum Keldysh Green's functions, respectively, for non-interacting systems. We apply the method to explore the connection between operator and entanglement growth, and transport in non-interacting systems with quasiperiodic and random disorder, like the one- and two-dimensional Aubry-André models and the two-dimensional Anderson model. In particular, we show that the growth of subsystem operator Rényi entropy and state von Neumann and Rényi entanglement entropies can capture both ballistic and sub-ballistic transport behaviors, like diffusive and anomalous diffusive transport, as well as localization in these systems.