Growth of R\'enyi Entropies in Interacting Integrable Models and the Breakdown of the Quasiparticle Picture

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Growth of Rényi Entropies in Interacting Integrable Models and the Breakdown of the Quasiparticle Picture

Bruno Bertini, Katja Klobas, Vincenzo Alba, Gianluca Lagnese, and Pasquale Calabrese

Phys. Rev. X 12, 031016 – Published 25 July 2022

Abstract

Rényi entropies are conceptually valuable and experimentally relevant generalizations of the celebrated von Neumann entanglement entropy. After a quantum quench in a clean quantum many-body system they generically display a universal linear growth in time followed by saturation. While a finite subsystem is essentially at local equilibrium when the entanglement saturates, it is genuinely out of equilibrium in the growth phase. In particular, the slope of the growth carries vital information on the nature of the system’s dynamics, and its characterization is a key objective of current research. Here we show that the slope of Rényi entropies can be determined by means of a spacetime duality transformation. In essence, we argue that the slope coincides with the stationary density of entropy of the model obtained by exchanging the roles of space and time. Therefore, very surprisingly, the slope of the entanglement is expressed as an equilibrium quantity. We use this observation to find an explicit exact formula for the slope of Rényi entropies in all integrable models treatable by thermodynamic Bethe ansatz and evolving from integrable initial states. Interestingly, this formula can be understood in terms of a quasiparticle picture only in the von Neumann limit.

Received 10 April 2022
Revised 9 June 2022
Accepted 13 June 2022

DOI:https://doi.org/10.1103/PhysRevX.12.031016

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Entanglement entropy Quantum entanglement Quantum quench

Quantum many-body systems

Duality Integrable systems

Quantum Information, Science & TechnologyStatistical Physics & Thermodynamics

Authors & Affiliations

Bruno Bertini ^1,2, Katja Klobas ³, Vincenzo Alba ⁴, Gianluca Lagnese ⁵, and Pasquale Calabrese^5,6

¹School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, United Kingdom
²Centre for the Mathematics and Theoretical Physics of Quantum Non-Equilibrium Systems, University of Nottingham, Nottingham NG7 2RD, United Kingdom
³Rudolf Peierls Centre for Theoretical Physics, Clarendon Laboratory, Oxford OX1 3PU, United Kingdom
⁴Dipartimento di Fisica, Università di Pisa, and INFN Sezione di Pisa, Largo Bruno Pontecorvo 3, Pisa, Italy
⁵SISSA and INFN Sezione di Trieste, via Bonomea 265, 34136 Trieste, Italy
⁶International Centre for Theoretical Physics (ICTP), Strada Costiera 11, 34151 Trieste, Italy

Popular Summary

The laws of thermodynamics stipulate that the entropy of a closed system cannot increase. Nevertheless, all the parts of a large quantum-mechanical system out of equilibrium experience an increase in their entropy. This surprising phenomenon is due to the growth of quantum correlations, or entanglement, among different subsystems. One remarkable, recently discovered law of nature says that the entanglement entropy—a standard quantifier of entanglement—is turned into thermodynamic entropy over time. Here, we study how one family of generalizations of the entanglement entropy, called Rényi entropies, changes over time.

The Rényi entropies are entanglement “monotones”—giving an estimate of the entanglement growth—with applications ranging from black holes to ultracold atoms. We mathematically demonstrate that their initial growth can be determined by a spacetime duality transformation. Namely, the rate of increase of Rényi entropies coincides with the stationary density of entropy in the model obtained by exchanging the roles of space and time. Thus, astonishingly, the nonequilibrium growth of the entanglement can be interpreted as an equilibrium quantity.

This unexpected result represents another fundamental step in the comprehension of nonequilibrium quantum many-body dynamics.

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Vol. 12, Iss. 3 — July - September 2022

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Figure 1
Diagrammatic representation of the reduced density matrix. The reduced density matrix $ρ_{A}$ is obtained by evolving an initial state in time and tracing over the complement $\bar{A}$ of the subsystem $A$ (graphically denoted by connecting the top and bottom legs). Alternatively, we can understand it as a trace of a product of powers of the space transfer matrix $W_{t}$ and the transfer matrix $W_{A, t}$ (both shaded in gray), as given by Eq. (8). In the limit $L \to \infty$ , the section of the tensor network corresponding to the rest of the system can be replaced by fixed points $⟨ M_{L, t} |$ and $| M_{R, t} ⟩$ of the space transfer matrix $W_{t}$ [cf. Eq. (10)].
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Figure 2
Schematic illustration of $tr [ρ_{A}^{3}]$ . Green and red rectangles are condensed representation of the green and red half of the time evolution from Fig. 1. The white connections on the left represent trace over the rest of the system, while the connections on the right correspond to matrix products and then an overall trace. In the limit of large $L - | A |$ the left and right parts of $\bar{A}$ can be substituted by fixed points $M_{L, t}^{*}$ and $M_{R, t}$ connecting pairs of time sheets. Analogously, when the subsystem size becomes large, the section in the middle can be replaced with fixed points $M_{L, t}^{†}$ , and $M_{R, t}^{T}$ . Note the additional transpose of fixed points, which is the consequence of connecting the opposite parity of pairs of neighbors. Thus we obtain the rightmost diagram, which corresponds precisely to the rhs of Eq. (13).
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Figure 3
Schematic representation of time evolution of Rule 54. In each time step the three-site gates $U$ are applied either to even ( $U_{L}^{e}$ ) or odd ( $U_{L}^{o}$ ) triplets of sites. Note that gates that overlap on at most one site commute.
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Figure 4
Numerical results for instantaneous Rényi slope $s_{α} (t)$ after the quench from the Néel state (80). The dashed lines represent the numerical data, with different colors distinguishing between different values of $α$ , while horizontal solid lines denote the analytical predictions for $s_{α}$ . Each panel corresponds to a different value of the anisotropy parameter: (a) $Δ = 2$ , (b) $Δ = 3$ , (c) $Δ = 4$ , and (d) $Δ = 6$ .
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Figure 5
Dynamics of the instantaneous Rényi slope $s_{α} (t)$ after the quench from the Majumdar-Ghosh state (81). The dashed lines represent the numerical data, and solid lines denote the analytical predictions for the asymptotic slope $s_{α}$ . Different colors distinguish between different $α$ , while the value of the anisotropy parameter is $Δ = 3$ in (a) and $Δ = 4$ in (b).
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Figure 6
Tensor network simulations of the entanglement dynamics for Rule 54. We consider an open system of $L = 50$ sites and we focus on the subsystem $A$ made of the first $ℓ$ sites, with $ℓ = 3, \dots, 13$ , represented by solid lines with different colors (see the legend). We fix the bond dimension equal to 4096. The dashed line is the exact prediction for $s_{α} t$ [note the absence of the factor of 2 with respect to Eq. (90) due to the choice of different boundary conditions], while the dot-dashed line represents the asymptotic thermodynamic entropy $d_{α} ℓ$ . Our conjecture (90) corresponds to the joining of the two straight lines. The entanglement velocity that rescales the time is $v_{α} = s_{α} / d_{α}$ .
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Physical Review X

Growth of Rényi Entropies in Interacting Integrable Models and the Breakdown of the Quasiparticle Picture

Bruno Bertini, Katja Klobas, Vincenzo Alba, Gianluca Lagnese, and Pasquale Calabrese

Phys. Rev. X 12, 031016 – Published 25 July 2022

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