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Journal of High Energy Physics

, 2016:109 | Cite as

State-dependent divergences in the entanglement entropy

  • Donald Marolf
  • Aron C. Wall
Open Access
Regular Article - Theoretical Physics

Abstract

We show the entanglement entropy in certain quantum field theories to contain state-dependent divergences. Both perturbative and holographic examples are exhibited. However, quantities such as the relative entropy and the generalized entropy of black holes remain finite, due to cancellation of divergences. We classify all possible state-dependent entanglement entropy divergences that can appear in both perturbatively renormalizeable and holographic covariant d ≤ 6 quantum field theories.

Keywords

Renormalization Regularization and Renormalons AdS-CFT Correspondence 

Notes

Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

References

  1. [1]
    A. Wehrl, General properties of entropy, Rev. Mod. Phys. 50 (1978) 221 [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  2. [2]
    H. Casini and M. Huerta, A finite entanglement entropy and the c-theorem, Phys. Lett. B 600 (2004) 142 [hep-th/0405111] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    H. Casini and M. Huerta, A c-theorem for the entanglement entropy, J. Phys. A 40 (2007) 7031 [cond-mat/0610375] [INSPIRE].
  4. [4]
    H. Casini and M. Huerta, On the RG running of the entanglement entropy of a circle, Phys. Rev. D 85 (2012) 125016 [arXiv:1202.5650] [INSPIRE].ADSGoogle Scholar
  5. [5]
    T. Grover, Entanglement Monotonicity and the Stability of Gauge Theories in Three Spacetime Dimensions, Phys. Rev. Lett. 112 (2014) 151601 [arXiv:1211.1392] [INSPIRE].ADSCrossRefGoogle Scholar
  6. [6]
    S.N. Solodukhin, The a-theorem and entanglement entropy, arXiv:1304.4411 [INSPIRE].
  7. [7]
    A. Kitaev and J. Preskill, Topological entanglement entropy, Phys. Rev. Lett. 96 (2006) 110404 [hep-th/0510092] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  8. [8]
    M. Levin and X.-G. Wen, Detecting Topological Order in a Ground State Wave Function, Phys. Rev. Lett. 96 (2006) 110405 [cond-mat/0510613].
  9. [9]
    T. Grover, Y. Zhang and A. Vishwanath, Entanglement Entropy as a Portal to the Physics of Quantum Spin Liquids, New J. Phys. 15 (2013) 025002 [arXiv:1302.0899] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  10. [10]
    S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    S. Ryu and T. Takayanagi, Aspects of Holographic Entanglement Entropy, JHEP 08 (2006) 045 [hep-th/0605073] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  12. [12]
    V.E. Hubeny, M. Rangamani and T. Takayanagi, A covariant holographic entanglement entropy proposal, JHEP 07 (2007) 062 [arXiv:0705.0016] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  13. [13]
    P.V. Buividovich and M.I. Polikarpov, Entanglement entropy in gauge theories and the holographic principle for electric strings, Phys. Lett. B 670 (2008) 141 [arXiv:0806.3376] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    W. Donnelly, Decomposition of entanglement entropy in lattice gauge theory, Phys. Rev. D 85 (2012) 085004 [arXiv:1109.0036] [INSPIRE].ADSGoogle Scholar
  15. [15]
    H. Casini, M. Huerta and J.A. Rosabal, Remarks on entanglement entropy for gauge fields, Phys. Rev. D 89 (2014) 085012 [arXiv:1312.1183] [INSPIRE].ADSGoogle Scholar
  16. [16]
    G. ’t Hooft, On the Quantum Structure of a Black Hole, Nucl. Phys. B 256 (1985) 727 [INSPIRE].
  17. [17]
    J.-G. Demers, R. Lafrance and R.C. Myers, Black hole entropy without brick walls, Phys. Rev. D 52 (1995) 2245 [gr-qc/9503003] [INSPIRE].
  18. [18]
    S.N. Solodukhin, Entanglement entropy of black holes, Living Rev. Rel. 14 (2011) 8 [arXiv:1104.3712] [INSPIRE].MATHGoogle Scholar
  19. [19]
    N. Iqbal and A.C. Wall, Anomalies of the Entanglement Entropy in Chiral Theories, arXiv:1509.04325 [INSPIRE].
  20. [20]
    W. Donnelly and A.C. Wall, Universality and double log terms in the entanglement entropy, forthcoming.Google Scholar
  21. [21]
    H. Casini, Mutual information challenges entropy bounds, Class. Quant. Grav. 24 (2007) 1293 [gr-qc/0609126] [INSPIRE].
  22. [22]
    H. Araki, Relative entropy of states of von neumann algebras, Publ. Res. Inst. Math. Sci. 11 (1976) 809.MathSciNetCrossRefMATHGoogle Scholar
  23. [23]
    R. Bousso, Z. Fisher, S. Leichenauer and A.C. Wall, Quantum focusing conjecture, Phys. Rev. D 93 (2016) 064044 [arXiv:1506.02669] [INSPIRE].ADSMathSciNetGoogle Scholar
  24. [24]
    A.C. Wall, The Generalized Second Law implies a Quantum Singularity Theorem, Class. Quant. Grav. 30 (2013) 165003 [Erratum ibid. 30 (2013) 199501] [arXiv:1010.5513] [INSPIRE].
  25. [25]
    R.C. Myers, R. Pourhasan and M. Smolkin, On Spacetime Entanglement, JHEP 06 (2013) 013 [arXiv:1304.2030] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  26. [26]
    N. Engelhardt and A.C. Wall, Quantum Extremal Surfaces: Holographic Entanglement Entropy beyond the Classical Regime, JHEP 01 (2015) 073 [arXiv:1408.3203] [INSPIRE].ADSCrossRefGoogle Scholar
  27. [27]
    C. Holzhey, F. Larsen and F. Wilczek, Geometric and renormalized entropy in conformal field theory, Nucl. Phys. B 424 (1994) 443 [hep-th/9403108] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  28. [28]
    D. Marolf, D. Minic and S.F. Ross, Notes on space-time thermodynamics and the observer dependence of entropy, Phys. Rev. D 69 (2004) 064006 [hep-th/0310022] [INSPIRE].ADSMathSciNetGoogle Scholar
  29. [29]
    H. Casini, Relative entropy and the Bekenstein bound, Class. Quant. Grav. 25 (2008) 205021 [arXiv:0804.2182] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  30. [30]
    R. Bousso, H. Casini, Z. Fisher and J. Maldacena, Proof of a Quantum Bousso Bound, Phys. Rev. D 90 (2014) 044002 [arXiv:1404.5635] [INSPIRE].ADSGoogle Scholar
  31. [31]
    R. Bousso, H. Casini, Z. Fisher and J. Maldacena, Entropy on a null surface for interacting quantum field theories and the Bousso bound, Phys. Rev. D 91 (2015) 084030 [arXiv:1406.4545] [INSPIRE].ADSMathSciNetGoogle Scholar
  32. [32]
    R.M. Wald, Black hole entropy is the Noether charge, Phys. Rev. D 48 (1993) R3427 [gr-qc/9307038] [INSPIRE].
  33. [33]
    T. Jacobson, G. Kang and R.C. Myers, On black hole entropy, Phys. Rev. D 49 (1994) 6587 [gr-qc/9312023] [INSPIRE].
  34. [34]
    V. Iyer and R.M. Wald, A Comparison of Noether charge and Euclidean methods for computing the entropy of stationary black holes, Phys. Rev. D 52 (1995) 4430 [gr-qc/9503052] [INSPIRE].
  35. [35]
    T. Faulkner, A. Lewkowycz and J. Maldacena, Quantum corrections to holographic entanglement entropy, JHEP 11 (2013) 074 [arXiv:1307.2892] [INSPIRE].ADSCrossRefGoogle Scholar
  36. [36]
    P. Calabrese and J. Cardy, Entanglement entropy and conformal field theory, J. Phys. A 42 (2009) 504005 [arXiv:0905.4013] [INSPIRE].MathSciNetMATHGoogle Scholar
  37. [37]
    F. Larsen and F. Wilczek, Renormalization of black hole entropy and of the gravitational coupling constant, Nucl. Phys. B 458 (1996) 249 [hep-th/9506066] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  38. [38]
    D.V. Fursaev, A. Patrushev and S.N. Solodukhin, Distributional Geometry of Squashed Cones, Phys. Rev. D 88 (2013) 044054 [arXiv:1306.4000] [INSPIRE].ADSGoogle Scholar
  39. [39]
    X. Dong, Holographic Entanglement Entropy for General Higher Derivative Gravity, JHEP 01 (2014) 044 [arXiv:1310.5713] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  40. [40]
    J. Camps, Generalized entropy and higher derivative Gravity, JHEP 03 (2014) 070 [arXiv:1310.6659] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  41. [41]
    J.H. Cooperman and M.A. Luty, Renormalization of Entanglement Entropy and the Gravitational Effective Action, JHEP 12 (2014) 045 [arXiv:1302.1878] [INSPIRE].ADSCrossRefGoogle Scholar
  42. [42]
    B.S. Kay and R.M. Wald, Theorems on the Uniqueness and Thermal Properties of Stationary, Nonsingular, Quasifree States on Space-Times with a Bifurcate Killing Horizon, Phys. Rept. 207 (1991) 49 [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  43. [43]
    R.M. Wald, Quantum field theory in curved spacetime and black hole thermodynamics, University of Chicago Press (1994).Google Scholar
  44. [44]
    C.R. Graham and E. Witten, Conformal anomaly of submanifold observables in AdS/CFT correspondence, Nucl. Phys. B 546 (1999) 52 [hep-th/9901021] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  45. [45]
    B.C. van Rees, Holographic renormalization for irrelevant operators and multi-trace counterterms, JHEP 08 (2011) 093 [arXiv:1102.2239] [INSPIRE].CrossRefMATHGoogle Scholar
  46. [46]
    B.C. van Rees, Irrelevant deformations and the holographic Callan-Symanzik equation, JHEP 10 (2011) 067 [arXiv:1105.5396] [INSPIRE].CrossRefMATHGoogle Scholar
  47. [47]
    V. Rosenhaus and M. Smolkin, Entanglement entropy, planar surfaces and spectral functions, JHEP 09 (2014) 119 [arXiv:1407.2891] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  48. [48]
    W. Donnelly and A.C. Wall, Do gauge fields really contribute negatively to black hole entropy?, Phys. Rev. D 86 (2012) 064042 [arXiv:1206.5831] [INSPIRE].ADSGoogle Scholar
  49. [49]
    M. Taylor and W. Woodhead, Renormalized entanglement entropy, JHEP 08 (2016) 165 [arXiv:1604.06808] [INSPIRE].ADSCrossRefGoogle Scholar
  50. [50]
    A. Lewkowycz and J. Maldacena, Generalized gravitational entropy, JHEP 08 (2013) 090 [arXiv:1304.4926] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  51. [51]
    T. Andrade and D. Marolf, AdS/CFT beyond the unitarity bound, JHEP 01 (2012) 049 [arXiv:1105.6337] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  52. [52]
    S. Minwalla, Restrictions imposed by superconformal invariance on quantum field theories, Adv. Theor. Math. Phys. 2 (1998) 781 [hep-th/9712074] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  53. [53]
    L.-Y. Hung, R.C. Myers and M. Smolkin, Some Calculable Contributions to Holographic Entanglement Entropy, JHEP 08 (2011) 039 [arXiv:1105.6055] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  54. [54]
    A.J. Amsel and D. Marolf, Energy Bounds in Designer Gravity, Phys. Rev. D 74 (2006) 064006 [Erratum ibid. D 75 (2007) 029901] [hep-th/0605101] [INSPIRE].
  55. [55]
    A.J. Amsel, T. Hertog, S. Hollands and D. Marolf, A Tale of two superpotentials: Stability and instability in designer gravity, Phys. Rev. D 75 (2007) 084008 [Erratum ibid. D 77 (2008) 049903] [hep-th/0701038] [INSPIRE].
  56. [56]
    D. Marolf and S.F. Ross, Boundary Conditions and New Dualities: Vector Fields in AdS/CFT, JHEP 11 (2006) 085 [hep-th/0606113] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  57. [57]
    P. Minces and V.O. Rivelles, Energy and the AdS/CFT correspondence, JHEP 12 (2001) 010 [hep-th/0110189] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  58. [58]
    W. Mueck, An improved correspondence formula for AdS/CFT with multitrace operators, Phys. Lett. B 531 (2002) 301 [hep-th/0201100] [INSPIRE].ADSCrossRefGoogle Scholar
  59. [59]
    P. Minces, Multitrace operators and the generalized AdS/CFT prescription, Phys. Rev. D 68 (2003) 024027 [hep-th/0201172] [INSPIRE].ADSGoogle Scholar
  60. [60]
    A. Sever and A. Shomer, A note on multitrace deformations and AdS/CFT, JHEP 07 (2002) 027 [hep-th/0203168] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  61. [61]
    S. Hollands, A. Ishibashi and D. Marolf, Counter-term charges generate bulk symmetries, Phys. Rev. D 72 (2005) 104025 [hep-th/0503105] [INSPIRE].ADSMathSciNetGoogle Scholar
  62. [62]
    H.J. Kim, L.J. Romans and P. van Nieuwenhuizen, The Mass Spectrum of Chiral N = 2 D=10 Supergravity on S 5,Phys. Rev. D 32 (1985) 389 [INSPIRE].ADSGoogle Scholar
  63. [63]
    A.C. Wall, A proof of the generalized second law for rapidly-evolving Rindler horizons, Phys. Rev. D 82 (2010) 124019 [arXiv:1007.1493] [INSPIRE].ADSGoogle Scholar
  64. [64]
    A.C. Wall, A proof of the generalized second law for rapidly changing fields and arbitrary horizon slices, Phys. Rev. D 85 (2012) 104049 [arXiv:1105.3445] [INSPIRE].ADSGoogle Scholar
  65. [65]
    D.L. Jafferis, A. Lewkowycz, J. Maldacena and S.J. Suh, Relative entropy equals bulk relative entropy, JHEP 06 (2016) 004 [arXiv:1512.06431] [INSPIRE].ADSCrossRefGoogle Scholar
  66. [66]
    X. Dong, D. Harlow and A.C. Wall, Reconstruction of Bulk Operators within the Entanglement Wedge in Gauge-Gravity Duality, Phys. Rev. Lett. 117 (2016) 021601 [arXiv:1601.05416] [INSPIRE].ADSCrossRefGoogle Scholar
  67. [67]
    S.N. Solodukhin, Nonminimal coupling and quantum entropy of black hole, Phys. Rev. D 56 (1997) 4968 [hep-th/9612061] [INSPIRE].ADSMathSciNetGoogle Scholar
  68. [68]
    M. Hotta, T. Kato and K. Nagata, A Comment on geometric entropy and conical space, Class. Quant. Grav. 14 (1997) 1917 [gr-qc/9611058] [INSPIRE].
  69. [69]
    I.R. Klebanov, T. Nishioka, S.S. Pufu and B.R. Safdi, Is Renormalized Entanglement Entropy Stationary at RG Fixed Points?, JHEP 10 (2012) 058 [arXiv:1207.3360] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  70. [70]
    T. Nishioka, Relevant Perturbation of Entanglement Entropy and Stationarity, Phys. Rev. D 90 (2014) 045006 [arXiv:1405.3650] [INSPIRE].ADSGoogle Scholar
  71. [71]
    J. Lee, A. Lewkowycz, E. Perlmutter and B.R. Safdi, Rényi entropy, stationarity and entanglement of the conformal scalar, JHEP 03 (2015) 075 [arXiv:1407.7816] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  72. [72]
    C.P. Herzog, Universal Thermal Corrections to Entanglement Entropy for Conformal Field Theories on Spheres, JHEP 10 (2014) 28 [arXiv:1407.1358] [INSPIRE].ADSCrossRefGoogle Scholar
  73. [73]
    J.S. Dowker, Expansion of Rényi entropy for free scalar fields, arXiv:1408.4055 [INSPIRE].
  74. [74]
    H. Casini, F.D. Mazzitelli and E. Testé, Area terms in entanglement entropy, Phys. Rev. D 91 (2015) 104035 [arXiv:1412.6522] [INSPIRE].ADSMathSciNetGoogle Scholar
  75. [75]
    V. Rosenhaus and M. Smolkin, Entanglement Entropy for Relevant and Geometric Perturbations, JHEP 02 (2015) 015 [arXiv:1410.6530] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  76. [76]
    D.V. Fursaev, Energy, Hamiltonian, Noether charge and black holes, Phys. Rev. D 59 (1999) 064020 [hep-th/9809049] [INSPIRE].ADSMathSciNetGoogle Scholar

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© The Author(s) 2016

Authors and Affiliations

  1. 1.Physics DepartmentUniversity of California, Santa BarbaraSanta BarbaraU.S.A.
  2. 2.School of Natural SciencesInstitute for Advanced StudyPrincetonU.S.A.

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