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Cited by 137Quantum many-body systemsBlack Holes and Theoretical PhysicsAlgebraic structures and combinatorial models

Thermalization and Revivals after a Quantum Quench in Conformal Field Theory

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Cardy, John·2014·Physical Review Letters
DOI 10.1103/physrevlett.112.220401
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Abstract

We consider a quantum quench in a finite system of length $L$ described by a $1+1$-dimensional conformal field theory (CFT), of central charge $c$, from a state with finite energy density corresponding to an inverse temperature $\ensuremath{\beta}\ensuremath{\ll}L$. For times $t$ such that $\ensuremath{\ell}/2<t<(L\ensuremath{-}\ensuremath{\ell})/2$ the reduced density matrix of a subsystem of length $\ensuremath{\ell}$ is exponentially close to a thermal density matrix. We compute exactly the overlap $\mathcal{F}$ of the state at time $t$ with the initial state and show that in general it is exponentially suppressed at large $L/\ensuremath{\beta}$. However, for minimal models with $c<1$ (more generally, rational CFTs), at times which are integer multiples of $L/2$ (for periodic boundary conditions, $L$ for open boundary conditions) there are (in general, partial) revivals at which $\mathcal{F}$ is $O(1)$, leading to an eventual complete revival with $\mathcal{F}=1$. There is also interesting structure at all rational values of $t/L$, related to properties of the CFT under modular transformations. At early times $t\ensuremath{\ll}(L\ensuremath{\beta}{)}^{1/2}$ there is a universal decay $\mathcal{F}\ensuremath{\sim}\mathrm{exp}(\ensuremath{-}(\ensuremath{\pi}c/3)L{t}^{2}/\ensuremath{\beta}({\ensuremath{\beta}}^{2}+4{t}^{2}))$. The effect of an irrelevant nonintegrable perturbation of the CFT is to progressively broaden each revival at $t=nL/2$ by an amount $O({n}^{1/2})$.

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  • The Mandelstam-Tamm bound can be used to derive a lower bound on the survival probability by an exponential function.

    “As was shown by Bhattacharyya [13], the Mandelstam-Tamm bound can be used to derive to lower bound the survival probability by an exponential function.”

    From Scrambling the spectral form factor: Unitarity constraints and exact results (2017) · cites this paper as [13]

  • Quantum speed limits set an upper bound to the rate at which a quantum system can evolve and can be used to analyze the scrambling of information.

    “Quantum speed limits set an upper bound to the rate at which a quantum system can evolve and as such can be used to analyze the scrambling of information.”

    From Scrambling the spectral form factor: Unitarity constraints and exact results (2017) · cites this paper as [13]

  • The bound derived in [13] holds for an initial pure state that undergoes unitary evolution.

    “The bound derived in [13] holds for an initial pure state that undergoes unitary evolution.”

    From Scrambling the spectral form factor: Unitarity constraints and exact results (2017) · cites this paper as [13]

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  1. [3]U.H. Danielsson, E. Keski-Vakkuri, and M. Kruczenski, JHEP 0002:039 (2000); S. Bhattacharyya and S. Minwalla, JHEP 0909 , 034 (2009); R. A. Janik and R. B. Peschanski, Phys. Rev. D 74 , 046007 (2006); H. Ebrahim and M. Headrick, arXiv:1010.5443 [hepth]; J. Abajo-Arrastia, J. Aparicio and E. Lopez, JHEP 1011 , 149 (2010); V. Balasubramanian et al. , Phys. Rev. Lett. 106 , 191601 (2011); A. Buchel, R.C. Myers, A. van Niekerk, Phys. Rev. Lett. 111 , 201602 (2013). · arXiv
  2. [6]M. Heyl, A. Polkovnikov and S. Kehrein, Phys. Rev. Lett. 110 , 135704 (2013); C. Karrasch and D. Schuricht, Phys. Rev. B 87 ,195104 (2013); M. Fagotti, arXiv:1308.0277; B. Pozsgay, arXiv:1308.3087; F. Andraschko and J. Sirker, arXiv:1312.4165; S. Vajna and B. D´ ora, arXiv:1401.2865. · arXiv

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Authors
Cardy, J.
Affiliations
University of Oxford (GB)
Corresponding author
Cardy, John
ORCID
Cardy, John
Venue
Physical Review Letters
Publisher
American Physical Society (APS)
Published
2014-06-05
Volume
112
Issue
22
Pages
220401-220401
ISSN
  • 0031-9007 (ISSN-L)
  • 1079-7114
DOI
10.1103/physrevlett.112.220401
OpenAlex
W1978152918
Cited by
137
References
35
Citations by year
2026: 3 · 2025: 6 · 2024: 6 · 2023: 12 · 2022: 10 · 2021: 16 · 2020: 14 · 2019: 12 · 2018: 14 · 2017: 13 · 2016: 15 · 2015: 8 · 2014: 8
FWCI
7.30
Type
journal article
Language
en
Open access
Licence
link.aps.org/licenses/aps-default-license (vor)
Locations
  • Physical Review Letters (publishedVersion)
  • PubMed (publishedVersion)
  • arXiv (Cornell University) (submittedVersion)
  • arXiv (Cornell University)
Funders
National Science Foundation (NSF PHY11-25915)
Topics
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum many-body systems
  • Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Black Holes and Theoretical Physics
  • Physical Sciences › Mathematics › Geometry and Topology › Algebraic structures and combinatorial models
Keywords
quantum quench, thermalization, revival, conformal field theory, reduced density matrix, minimal models, rational conformal field theory, modular transformations, finite systems, state overlap, central charge

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