Skip to main content
SpringerLink
Log in

Holographic entanglement plateaux

  • Published:
Journal of High Energy Physics Aims and scope Submit manuscript

Abstract

We consider the entanglement entropy for holographic field theories in finite volume. We show that the Araki-Lieb inequality is saturated for large enough subregions, implying that the thermal entropy can be recovered from the knowledge of the region and its complement. We observe that this actually is forced upon us in holographic settings due to non-trivial features of the causal wedges associated with a given boundary region. In the process, we present an infinite set of extremal surfaces in Schwarzschild-AdS geometry anchored on a given entangling surface. We also offer some speculations regarding the homology constraint required for computing holographic entanglement entropy.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  2. S. Ryu and T. Takayanagi, Aspects of Holographic Entanglement Entropy, JHEP 08 (2006) 045 [hep-th/0605073] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  3. V.E. Hubeny, M. Rangamani and T. Takayanagi, A covariant holographic entanglement entropy proposal, JHEP 07 (2007) 062 [arXiv:0705.0016] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  4. R. Bousso, The holographic principle, Rev. Mod. Phys. 74 (2002) 825 [hep-th/0203101] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  5. D.V. Fursaev, Proof of the holographic formula for entanglement entropy, JHEP 09 (2006) 018 [hep-th/0606184] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  6. M. Headrick, Entanglement Renyi entropies in holographic theories, Phys. Rev. D 82 (2010) 126010 [arXiv:1006.0047] [INSPIRE].

    ADS  Google Scholar 

  7. H. Casini, M. Huerta and R.C. Myers, Towards a derivation of holographic entanglement entropy, JHEP 05 (2011) 036 [arXiv:1102.0440] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  8. A. Lewkowycz and J. Maldacena, Generalized gravitational entropy, arXiv:1304.4926 [INSPIRE].

  9. T. Hartman, Entanglement entropy at large central charge, arXiv:1303.6955 [INSPIRE].

  10. T. Faulkner, The entanglement Renyi entropies of disjoint intervals in AdS/CFT, arXiv:1303.7221 [INSPIRE].

  11. H. Araki and E. Lieb, Entropy inequalities, Commun. Math. Phys. 18 (1970) 160.

    Article  MathSciNet  ADS  Google Scholar 

  12. V.E. Hubeny and H. Maxfield, work in progress.

  13. T. Albash and C.V. Johnson, Holographic studies of entanglement entropy in superconductors, JHEP 05 (2012) 079 [arXiv:1202.2605] [INSPIRE].

    Article  ADS  Google Scholar 

  14. A. Belin, A. Maloney and S. Matsuura, Holographic phases of Renyi entropies, arXiv:1306.2640 [INSPIRE].

  15. H. Liu and M. Mezei, A refinement of entanglement entropy and the number of degrees of freedom, arXiv:1202.2070 [INSPIRE].

  16. M. Headrick and T. Takayanagi, A holographic proof of the strong subadditivity of entanglement entropy, Phys. Rev. D 76 (2007) 106013 [arXiv:0704.3719] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  17. T. Azeyanagi, T. Nishioka and T. Takayanagi, Near extremal black hole entropy as entanglement entropy via AdS 2 /CF T 1, Phys. Rev. D 77 (2008) 064005 [arXiv:0710.2956] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  18. D.D. Blanco, H. Casini, L.-Y. Hung and R.C. Myers, Relative entropy and holography, arXiv:1305.3182 [INSPIRE].

  19. B. Freivogel et al., Inflation in AdS/CFT, JHEP 03 (2006) 007 [hep-th/0510046] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  20. T. Nishioka and T. Takayanagi, AdS bubbles, entropy and closed string tachyons, JHEP 01 (2007) 090 [hep-th/0611035] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  21. I.R. Klebanov, D. Kutasov and A. Murugan, Entanglement as a probe of confinement, Nucl. Phys. B 796 (2008) 274 [arXiv:0709.2140] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  22. V.E. Hubeny and M. Rangamani, Causal holographic information, JHEP 06 (2012) 114 [arXiv:1204.1698] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  23. V.E. Hubeny, M. Rangamani and E. Tonni, Global properties of causal wedges in asymptotically AdS spacetimes, arXiv:1306.4324 [INSPIRE].

  24. P. Calabrese and J.L. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech. 0406 (2004) P06002 [hep-th/0405152] [INSPIRE].

    Article  Google Scholar 

  25. N. Ogawa, T. Takayanagi and T. Ugajin, Holographic Fermi surfaces and entanglement entropy, JHEP 01 (2012) 125 [arXiv:1111.1023] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  26. C.P. Herzog and M. Spillane, Tracing through scalar entanglement, Phys. Rev. D 87 (2013) 025012 [arXiv:1209.6368] [INSPIRE].

    ADS  Google Scholar 

  27. M.M. Caldarelli, O.J. Dias, R. Emparan and D. Klemm, Black holes as lumps of fluid, JHEP 04 (2009) 024 [arXiv:0811.2381] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  28. R. Emparan, R. Suzuki and K. Tanabe, The large D limit of general relativity, JHEP 06 (2013) 009 [arXiv:1302.6382] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  29. V.E. Hubeny, Extremal surfaces as bulk probes in AdS/CFT, JHEP 07 (2012) 093 [arXiv:1203.1044] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  30. J. Abajo-Arrastia, J. Aparicio and E. Lopez, Holographic evolution of entanglement entropy, JHEP 11 (2010) 149 [arXiv:1006.4090] [INSPIRE].

    Article  ADS  Google Scholar 

  31. T. Hartman and J. Maldacena, Time evolution of entanglement entropy from black hole interiors, JHEP 05 (2013) 014 [arXiv:1303.1080] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  32. H. Liu and S.J. Suh, Entanglement tsunami: universal scaling in holographic thermalization, arXiv:1305.7244 [INSPIRE].

  33. S.H. Shenker and D. Stanford, Black holes and the butterfly effect, arXiv:1306.0622 [INSPIRE].

  34. A.C. Wall, Maximin surfaces and the strong subadditivity of the covariant holographic entanglement entropy, arXiv:1211.3494 [INSPIRE].

  35. B. Czech, J.L. Karczmarek, F. Nogueira and M. Van Raamsdonk, The gravity dual of a density matrix, Class. Quant. Grav. 29 (2012) 155009 [arXiv:1204.1330] [INSPIRE].

    Article  ADS  Google Scholar 

  36. D. Marolf, Black holes, AdS and CFTs, Gen. Rel. Grav. 41 (2009) 903 [arXiv:0810.4886] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  37. E. Witten, Anti-de Sitter space, thermal phase transition and confinement in gauge theories, Adv. Theor. Math. Phys. 2 (1998) 505 [hep-th/9803131] [INSPIRE].

    MathSciNet  MATH  Google Scholar 

  38. B. Sundborg, The Hagedorn transition, deconfinement and N = 4 SYM theory, Nucl. Phys. B 573 (2000) 349 [hep-th/9908001] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  39. O. Aharony, J. Marsano, S. Minwalla, K. Papadodimas and M. Van Raamsdonk, The Hagedorn-deconfinement phase transition in weakly coupled large-N gauge theories, Adv. Theor. Math. Phys. 8 (2004) 603 [hep-th/0310285] [INSPIRE].

    Article  MathSciNet  MATH  Google Scholar 

  40. I.A. Morrison and M.M. Roberts, Mutual information between thermo-field doubles and disconnected holographic boundaries, arXiv:1211.2887 [INSPIRE].

  41. T. Takayanagi and T. Ugajin, Measuring black hole formations by entanglement entropy via coarse-graining, JHEP 11 (2010) 054 [arXiv:1008.3439] [INSPIRE].

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Centre for Particle Theory & Department of Mathematical Sciences, Science Laboratories,, South Road, Durham, DH1 3LE, U.K.

    Veronika E. Hubeny, Henry Maxfield & Mukund Rangamani

  2. SISSA and INFN, via Bonomea 265, 34136, Trieste, Italy

    Erik Tonni

Authors
  1. Veronika E. Hubeny
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Henry Maxfield
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mukund Rangamani
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Erik Tonni
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mukund Rangamani.

Additional information

ArXiv ePrint: 1306.4004

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hubeny, V.E., Maxfield, H., Rangamani, M. et al. Holographic entanglement plateaux. J. High Energ. Phys. 2013, 92 (2013). https://doi.org/10.1007/JHEP08(2013)092

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/JHEP08(2013)092

Keywords

Search

Navigation