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arXiv:2012.11917 · Communications Physics

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

Marko Žnidarič

Published May 10, 2021• 0 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1038/s42005-021-00595-3

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Comment on "Entanglement growth in diffusive systems"

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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.
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.
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.
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.
We study many-body localization for a disordered chain of spin-1/2 fermions. When both down and up components are exposed to the same strong disorder, a power-law growth of the entanglement entropy was observed [P. Prelov\ifmmode \check{s}\else \v{s}\fi{}ek et al., Phys. Rev. B 94, 241104 (2016)] that suggests that many-body localization is not complete; the density (charge) degree of freedom is localized, while the spin degree of freedom is apparently delocalized. We show that this powerlike behavior is only a transient effect and that, for longer times, the growth of the entanglement entropy is logarithmic in time. At the same time, the decay of the spin correlation slows down, showing that the spin transport is strongly reduced at long times. The dynamics of the system is quite similar to the one of many-body localized systems. Because of the limited duration of the numerical experiment, it is not possible to determine whether the system is truly many-body localized or diffusing very slowly at long times.
The large-scale behaviour of entanglement entropy in finite-density states, in and out of equilibrium, can be understood using the physical picture of particle pairs. However, the full theoretical origin of this picture is not fully established yet. In this work, we clarify this picture by investigating entanglement entropy using its connection with the large-deviation theory for thermodynamic and hydrodynamic fluctuations. We apply the universal framework of Ballistic Fluctuation Theory (BFT), based the Euler hydrodynamics of the model, to correlation functions of branch-point twist fields, the starting point for computing Rényi entanglement entropies within the replica approach. Focusing on free fermionic systems in order to illustrate the ideas, we show that both the equilibrium behavior and the dynamics of Rényi entanglement entropies can be fully derived from the BFT. In particular, we emphasise that long-range correlations develop after quantum quenches, and accounting for these explain the structure of the entanglement growth. We further show that this growth is related to fluctuations of charge transport, generalising to quantum quenches the relation between charge fluctuations and entanglement observed earlier. The general ideas we introduce suggest that the large-scale behaviour of entanglement has its origin within hydrodynamic fluctuations.