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  • Letter
  • Open Access

Subregion spectrum form factor via pseudoentropy

Kanato Goto1,*, Masahiro Nozaki1,2,†, and Kotaro Tamaoka3,‡

  • 1RIKEN Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS), Wako, Saitama 351-0198, Japan
  • 2Kavli Institute for Theoretical Sciences and CAS Center for Excellence in Topological Quantum Computation, University of Chinese Academy of Sciences, Beijing 100190, China
  • 3Department of Physics, College of Humanities and Sciences, Nihon University, Sakura-josui, Tokyo 156-8550, Japan

  • *kanato.goto@riken.jp
  • †masahiro.nozaki@riken.jp
  • ‡tamaoka.kotaro@nihon-u.ac.jp
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Phys. Rev. D 104, L121902 – Published 7 December, 2021

DOI: https://doi.org/10.1103/PhysRevD.104.L121902

Abstract

We introduce a subsystem generalization of the spectral form factor via pseudoentropy, the von-Neumann entropy for the reduced transition matrix. We consider a transition matrix between the thermofield double state and its time-evolved state in two-dimensional conformal field theories, and study the time dependence of the pseudoentropy for a single interval. We show that the real part of the pseudoentropy behaves similarly to the spectral form factor; it starts from the thermal entropy, initially drops to the minimum, then it starts increasing, and then finally it approaches the vacuum entanglement entropy. We also study the theory dependence of its behavior by considering theories on a compact space.

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References (36)

  1. J. Maldacena, The large-N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38, 1113 (1999).
  2. S. Ryu and T. Takayanagi, Holographic Derivation of Entanglement Entropy from the anti–de Sitter Space/Conformal Field Theory Correspondence, Phys. Rev. Lett. 96, 181602 (2006).
  3. S. Ryu and T. Takayanagi, Aspects of holographic entanglement entropy, J. High Energy Phys. 08 (2006) 045.
  4. P. Calabrese and J. Cardy, Evolution of entanglement entropy in one-dimensional systems, J. Stat. Mech. (2005) P04010.
  5. P. Calabrese and J. Cardy, Entanglement and correlation functions following a local quench: A conformal field theory approach, J. Stat. Mech. (2007) P10004.
  6. P. Hosur, X.-L. Qi, D. A. Roberts, and B. Yoshida, Chaos in quantum channels, J. High Energy Phys. 02 (2016) 004.
  7. L. Nie, M. Nozaki, S. Ryu, and M. T. Tan, Signature of quantum chaos in operator entanglement in 2d CFTs, J. Stat. Mech. (2019) 093107.
  8. J. Kudler-Flam, M. Nozaki, S. Ryu, and M. T. Tan, Quantum vs classical information: Operator negativity as a probe of scrambling, J. High Energy Phys. 01 (2020) 031.
  9. H. Li and F. D. M. Haldane, Entanglement Spectrum as a Generalization of Entanglement Entropy: Identification of Topological Order in Non-Abelian Fractional Quantum Hall Effect States, Phys. Rev. Lett. 101, 010504 (2008).
  10. F. Pollmann, A. M. Turner, E. Berg, and M. Oshikawa, Entanglement spectrum of a topological phase in one dimension, Phys. Rev. B 81, 064439 (2010).
  11. H. Casini, M. Huerta, and R. C. Myers, Towards a derivation of holographic entanglement entropy, J. High Energy Phys. 05 (2011) 036.
  12. J. S. Cotler, G. Gur-Ari, M. Hanada, J. Polchinski, P. Saad, S. H. Shenker, D. Stanford, A. Streicher, and M. Tezuka, Black holes and random matrices, J. High Energy Phys. 05 (2017) 118.
  13. K. Papadodimas and S. Raju, Local Operators in the Eternal Black Hole, Phys. Rev. Lett. 115, 211601 (2015).
  14. M. L. Mehta, Random Matrices (Elsevier, New York, 2004).
  15. Y. Nakata, T. Takayanagi, Y. Taki, K. Tamaoka, and Z. Wei, New holographic generalization of entanglement entropy, Phys. Rev. D 103, 026005 (2021).
  16. X. Chen and A. W. W. Ludwig, Universal spectral correlations in the chaotic wave function, and the development of quantum chaos, Phys. Rev. B 98, 064309 (2018).
  17. X. Chen and T. Zhou, Operator scrambling and quantum chaos (2018).
  18. C.-T. Ma and C.-H. Wu, Quantum entanglement and spectral form factor (2020).
  19. A. Mollabashi, N. Shiba, T. Takayanagi, K. Tamaoka, and Z. Wei, Pseudo Entropy in Free Quantum Field Theories, Phys. Rev. Lett. 126, 081601 (2021).
  20. A. Mollabashi, N. Shiba, T. Takayanagi, K. Tamaoka, and Z. Wei, Aspects of pseudo entropy in field theories (2021).
  21. T. Nishioka, T. Takayanagi, and Y. Taki, Topological pseudo entropy (2021).

  22. We read off the scaling of the Thouless time from the numerical plots of the real part of the pseudoentropy. Small oscillations are observed in the ramp region, and the Heisenberg time is defined as the time at which the pseudo entropy ceases to oscillate.

  23. A. Chan, R. M. Nandkishore, M. Pretko, and G. Smith, Unitary-projective entanglement dynamics, Phys. Rev. B 99, 224307 (2019).
  24. Y. Li, X. Chen, and M. P. A. Fisher, Quantum Zeno effect and the many-body entanglement transition, Phys. Rev. B 98, 205136 (2018).
  25. B. Skinner, J. Ruhman, and A. Nahum, Measurement-Induced Phase Transitions in the Dynamics of Entanglement, Phys. Rev. X 9, 031009 (2019).
  26. P. Calabrese and J. L. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech. (2004) P06002.

  27. These timescales are also clear from the equation (19) and (pseudo2): for the subsystem Thouless time, we can see the periodic oscillation becomes relevant when the real part of inside of sinh decreases to order 1. This matches t∼βℓ. For the subsystem Heisenberg time, we can see the periodic oscillation stops when t becomes order ℓ.

  28. P. Calabrese and A. Lefevre, Entanglement spectrum in one-dimensional systems, Phys. Rev. A 78, 032329 (2008).
  29. T. Azeyanagi, T. Nishioka, and T. Takayanagi, Near extremal black hole entropy as entanglement entropy via AdS2/CFT1, Phys. Rev. D 77, 064005 (2008).
  30. C. P. Herzog and T. Nishioka, Entanglement entropy of a massive fermion on a torus, J. High Energy Phys. 03 (2013) 077.
  31. M. Headrick and T. Takayanagi, A holographic proof of the strong subadditivity of entanglement entropy, Phys. Rev. D 76, 106013 (2007).
  32. C. T. Asplund, A. Bernamonti, F. Galli, and T. Hartman, Holographic entanglement entropy from 2d CFT: Heavy states and local quenches, J. High Energy Phys. 02 (2015) 171.
  33. P. Saad, S. H. Shenker, and D. Stanford, A semiclassical ramp in SYK and in gravity (2018).
  34. P. Saad, S. H. Shenker, and D. Stanford, JT gravity as a matrix integral (2019).
  35. G. Penington, S. H. Shenker, D. Stanford, and Z. Yang, Replica wormholes and the black hole interior (2019).
  36. P.-Y. Chang, J.-S. You, X. Wen, and S. Ryu, Entanglement spectrum and entropy in topological non-Hermitian systems and nonunitary conformal field theory, Phys. Rev. Research 2, 033069 (2020).

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Phys. Rev. D 104, L121902– Published 7 December, 2021
  • Received 8 September 2021
  • Accepted 15 November 2021
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