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  • Letter
  • Open Access

Time scaling of entanglement in integrable scale-invariant theories

M. Reza Mohammadi Mozaffar1,2 and Ali Mollabashi3,4

  • 1Department of Physics, University of Guilan, P.O. Box 41335-1914, Rasht, Iran
  • 2School of Physics, Institute for Research in Fundamental Sciences (IPM), 19538-33511 Tehran, Iran
  • 3Center for Gravitational Physics, Yukawa Institute for Theoretical Physics, Kyoto University, Kitashirakawa Oiwakecho, Sakyo-ku, Kyoto 606-8502, Japan
  • 4Max-Planck-Institut for Physics, Werner-Heisenberg-Institut, 80805 Munich, Germany

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Phys. Rev. Research 4, L022010 – Published 14 April, 2022

DOI: https://doi.org/10.1103/PhysRevResearch.4.L022010

Abstract

In two-dimensional isotropic scale-invariant theories, the time scaling of the entanglement entropy of a segment is fixed via the conformal symmetry. We consider scale invariance in a more general sense and show that in integrable theories in which the scale invariance is anisotropic between time and space, parametrized by z, most of the entanglement is carried by the slow modes. At early times entanglement grows linearly due to the contribution of the fast modes, before smoothly entering a slow mode regime where it grows forever with t11−z. The slow-mode regime admits a logarithmic enhancement in bosonic theories. We check our analytical results against numerical simulations in corresponding fermionic and bosonic lattice models and find extremely good agreement. We show that due to the dominance of the slow modes in these non-relativistic theories, local quantum information is scrambled independent of z in a stronger way, compared to their relativistic counterparts.

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References (33)

  1. A. Polkovnikov, K. Sengupta, A. Silva, and M. Vengalattore, Colloquium: Nonequilibrium dynamics of closed interacting quantum systems, Rev. Mod. Phys. 83, 863 (2011); C. Gogolin and J. Eisert, Equilibration, thermalisation, and the emergence of statistical mechanics in closed quantum systems, Rep. Prog. Phys. 79, 056001 (2016); P. Calabrese, Entanglement spreading in non-equilibrium integrable systems, SciPost Phys. Lect. Notes 20 (2020); S. D. Mathur, The Fuzzball proposal for black holes: An elementary review, Fortschr. Phys. 53, 793 (2005); J. Polchinski, The black hole information problem, in Proceedings of the Theoretical Advanced Study Institute in Elementary Particle Physics 2016: New Frontiers in Fields and Strings (World Scientific, 2017), pp. 353–397; A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian, and A. Tajdini, The entropy of Hawking radiation, Rev. Mod. Phys. 93, 035002 (2021).
  2. T. Kinoshita, T. Wenger, and D. S. Weiss, A quantum Newton's cradle, Nature (London) 440, 900 (2008); S. Hofferberth et al., Non-equilibrium coherence dynamics in one-dimensional Bose gases, ibid. 449, 324 (2007); S. Trotzky et al., Probing the relaxation towards equilibrium in an isolated strongly correlated 1D Bose gas, Nat. Phys. 8, 325 (2012); M. Cheneau et al., Light-cone-like spreading of correlations in a quantum many-body system, Nature (London) 481, 484 (2012); M. Gring et al., Relaxation dynamics and prethermalisation in an isolated quantum system, Science 337, 1318 (2012); T. Langen et al., Local emergence of thermal correlations in an isolated quantum many-body system, Nat. Phys. 9, 640 (2013); Experimental observation of a generalized Gibbs ensemble, Science 348, 207 (2015); A. M. Kaufman et al., Quantum thermalization through entanglement in an isolated many-body system, ibid. 353, 794 (2016); T. Brydges et al., Probing Rényi entanglement entropy via randomized measurements, ibid. 364, 260 (2019); A. Lukin et al., Probing entanglement in a many-body localized system, ibid. 364, 6437 (2019).
  3. P. Calabrese and J. L. Cardy, Time-Dependence of Correlation Functions Following a Quantum Quench, Phys. Rev. Lett. 96, 136801 (2006).
  4. P. Calabrese and J. L. Cardy, Evolution of entanglement entropy in one-dimensional systems, J. Stat. Mech. (2005) P04010.
  5. V. Alba and P. Calabrese, Entanglement and thermodynamics after a quantum quench in integrable systems, Proc. Natl. Acad. Sci. USA 114, 7947 (2017).
  6. Our analysis for z=1 recovers the well-known results for CFT.
  7. S. Sachdev, Quantum Phase Transitions (Cambridge University, Cambridge, England, 2011).
  8. V. Alba and P. Calabrese, Entanglement dynamics after quantum quenches in generic integrable systems, SciPost Phys. 4, 017 (2018).
  9. J. Alexandre, Lifshitz-type Quantum Field Theories in Particle Physics, Int. J. Mod. Phys. A 26, 4523 (2011).
  10. M. R. Mohammadi Mozaffar and A. Mollabashi (to be published).
  11. H. Casini, H. Liu, and M. Mezei, Spread of entanglement and causality, J. High Energy Phys. 07 (2016) 077.
  12. No Lieb-Robinson bound has been proven so far for these theories. In theories that such a bound with an exponential tail exists, it is known that the leakage of information outside of the effective light-cone is negligible [13].
  13. S. Bravyi, M. B. Hastings, and F. Verstraete, Lieb-Robinson Bounds and the Generation of Correlations and Topological Quantum Order, Phys. Rev. Lett. 97, 050401 (2006); J. Eisert and T. J. Osborne, General Entanglement Scaling Laws from Time Evolution, ibid. 97, 150404 (2006).
  14. J. S. Cotler, M. P. Hertzberg, M. Mezei, and M. T. Mueller, Entanglement growth after a global quench in free scalar field theory, J. High Energy Phys. 11 (2016) 166.
  15. Infinite saturation time may happen in non-scale-invariant relativistic theories as well, while the contribution of slow modes in nonrelativistic scale-invariant theories is dominant due to the relatively large value of s(k) for slow modes.
  16. Recently nonlinear growth of entanglement has been reported in inhomogeneous space-times as well [17].
  17. A. Kosior and M. Heyl, Nonlinear entanglement growth in inhomogeneous space-times, Phys. Rev. Research 2, 043036 (2020).
  18. I. Peschel, Calculation of reduced density matrices from correlation functions, J. Phys. A: Math. Gen. 36, L205 (2003); K. Audenaert, J. Eisert, M. B. Plenio, and R. F. Werner, Entanglement properties of the harmonic chain, Phys. Rev. A 66, 042327 (2002); G. Vidal, J. I. Latorre, E. Rico, and A. Kitaev, Entanglement in Quantum Critical Phenomena, Phys. Rev. Lett. 90, 227902 (2003); J. I. Latorre, E. Rico, and G. Vidal, Ground state entanglement in quantum spin chains, Quantum Inf. Comput. 4, 48 (2004); M. B. Plenio, J. Eisert, J. Dreissig, and M. Cramer, Entropy, Entanglement, and Area: Analytical Results for Harmonic Lattice Systems, Phys. Rev. Lett. 94, 060503 (2005); I. Peschel and V. Eisler, Reduced density matrices and entanglement entropy in free lattice models, J. Phys. A 42, 504003 (2009); H. Casini and M. Huerta, Entanglement entropy in free quantum field theory, ibid. 42, 504007 (2009).
  19. M. R. Mohammadi Mozaffar and A. Mollabashi, Entanglement in Lifshitz-type quantum field theories, J. High Energy Phys. 07 (2017) 120; T. He, J. M. Magan, and S. Vandoren, Entanglement entropy in Lifshitz theories, SciPost Phys. 3 (2017) 034; M. R. Mohammadi Mozaffar and A. Mollabashi, Logarithmic negativity in Lifshitz harmonic models, J. Stat. Mech. (2018) 053113; Entanglement evolution in Lifshitz-type scalar theories, J. High Energy Phys. 01 (2019) 137.
  20. See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevResearch.4.L022010 for the details of our free Lifshitz Dirac fermion and free Lifshitz scalar theories.
  21. A different fermionic model with similar properties has been introduced in Ref. [22].
  22. D. Hartmann, K. Kavanagh, and S. Vandoren, Entanglement entropy with Lifshitz fermions, SciPost Phys. 11, 031 (2021).
  23. V. Alba and P. Calabrese, Quantum information scrambling after a quantum quench, Phys. Rev. B 100, 115150 (2019).
  24. Note that here our focus is on the role of slow modes in scrambling of quantum information in integrable theories. For a more complete investigation of quantum information scrambling in such theories via a wider set of scrambling measures, see Refs. [10, 25].
  25. T. Kuwahara and K. Saito, Absence of Fast Scrambling in Thermodynamically Stable Long-Range Interacting Systems, Phys. Rev. Lett. 126, 030604 (2021).
  26. C. T. Asplund, A. Bernamonti, F. Galli, and T. Hartman, Entanglement scrambling in 2d conformal field theory, J. High Energy Phys. 09 (2015) 110; V. Balasubramanian, A. Bernamonti, N. Copland, B. Craps, and F. Galli, Thermalization of mutual and tripartite information in strongly coupled two-dimensional conformal field theories, Phys. Rev. D 84, 105017 (2011); C. T. Asplund and A. Bernamonti, Mutual information after a local quench in conformal field theory, ibid. 89, 066015 (2014); S. Leichenauer and M. Moosa, Entanglement tsunami in (1+1)-dimensions, ibid. 92, 126004 (2015).
  27. R. Modak, V. Alba, and P. Calabrese, Entanglement revivals as a probe of scrambling in finite quantum systems, J. Stat. Mech. (2020) 083110.
  28. P. Hauke, L. Tagliacozzo, Spread of Correlations in Long-Range Interacting Quantum Systems, Phys. Rev. Lett. 111, 207202 (2013).
  29. M. Foss-Feig, Z. X. Gong, C. W. Clark, and A. V. Gorshkov, Nearly Linear Light Cones in Long-Range Interacting Quantum Systems, Phys. Rev. Lett. 114, 157201 (2015); M. C. Tran, A. Y. Guo, Y. Su, J. R. Garrison, Z. Eldredge, M. Foss-Feig, A. M. Childs, and A. V. Gorshkov, Locality and Digital Quantum Simulation of Power-Law Interactions, Phys. Rev. X 9, 031006 (2019); T. Kuwahara and K. Saito, Strictly Linear Light Cones in Long-Range Interacting Systems of Arbitrary Dimensions, ibid. 10, 031010 (2020).
  30. P. Richerme et al., Non-local propagation of correlations in long-range interacting quantum systems, Nature (London) 511, 198 (2014); P. Jurcevic et al., Observation of entanglement propagation in a quantum many-body system, ibid. 511, 202 (2014).
  31. O. A. Castro-Alvaredo, M. Lencsés, I. M. Szécsényi, and J. Viti, Entanglement Oscillations near a Quantum Critical Point, Phys. Rev. Lett. 124, 230601 (2020).
  32. V. Keranen, E. Keski-Vakkuri, and L. Thorlacius, Thermalization and entanglement following a non-relativistic holographic quench, Phys. Rev. D 85, 026005 (2012); T. Hartman and J. Maldacena, Time evolution of entanglement entropy from black hole interiors, J. High Energy Phys. 05 (2013) 014; M. Alishahiha, A. Faraji Astaneh, and M. R. Mohammadi Mozaffar, Thermalization in backgrounds with hyperscaling violating factor, Phys. Rev. D 90, 046004 (2014); P. Fonda, L. Franti, V. Keränen, E. Keski-Vakkuri, L. Thorlacius, and E. Tonni, Holographic thermalization with Lifshitz scaling and hyperscaling violation, J. High Energy Phys. 08 (2014) 051.
  33. https://turin.ipm.ir/.

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Phys. Rev. Research 4, L022010– Published 14 April, 2022
  • Received 6 July 2021
  • Accepted 11 March 2022
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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. Open access publication funded by the Max Planck Society.

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