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.
Article PDF
Similar content being viewed by others
References
A. Wehrl, General properties of entropy, Rev. Mod. Phys. 50 (1978) 221 [INSPIRE].
H. Casini and M. Huerta, A finite entanglement entropy and the c-theorem, Phys. Lett. B 600 (2004) 142 [hep-th/0405111] [INSPIRE].
H. Casini and M. Huerta, A c-theorem for the entanglement entropy, J. Phys. A 40 (2007) 7031 [cond-mat/0610375] [INSPIRE].
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].
T. Grover, Entanglement Monotonicity and the Stability of Gauge Theories in Three Spacetime Dimensions, Phys. Rev. Lett. 112 (2014) 151601 [arXiv:1211.1392] [INSPIRE].
S.N. Solodukhin, The a-theorem and entanglement entropy, arXiv:1304.4411 [INSPIRE].
A. Kitaev and J. Preskill, Topological entanglement entropy, Phys. Rev. Lett. 96 (2006) 110404 [hep-th/0510092] [INSPIRE].
M. Levin and X.-G. Wen, Detecting Topological Order in a Ground State Wave Function, Phys. Rev. Lett. 96 (2006) 110405 [cond-mat/0510613].
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].
S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].
S. Ryu and T. Takayanagi, Aspects of Holographic Entanglement Entropy, JHEP 08 (2006) 045 [hep-th/0605073] [INSPIRE].
V.E. Hubeny, M. Rangamani and T. Takayanagi, A covariant holographic entanglement entropy proposal, JHEP 07 (2007) 062 [arXiv:0705.0016] [INSPIRE].
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].
W. Donnelly, Decomposition of entanglement entropy in lattice gauge theory, Phys. Rev. D 85 (2012) 085004 [arXiv:1109.0036] [INSPIRE].
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].
G. ’t Hooft, On the Quantum Structure of a Black Hole, Nucl. Phys. B 256 (1985) 727 [INSPIRE].
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].
S.N. Solodukhin, Entanglement entropy of black holes, Living Rev. Rel. 14 (2011) 8 [arXiv:1104.3712] [INSPIRE].
N. Iqbal and A.C. Wall, Anomalies of the Entanglement Entropy in Chiral Theories, arXiv:1509.04325 [INSPIRE].
W. Donnelly and A.C. Wall, Universality and double log terms in the entanglement entropy, forthcoming.
H. Casini, Mutual information challenges entropy bounds, Class. Quant. Grav. 24 (2007) 1293 [gr-qc/0609126] [INSPIRE].
H. Araki, Relative entropy of states of von neumann algebras, Publ. Res. Inst. Math. Sci. 11 (1976) 809.
R. Bousso, Z. Fisher, S. Leichenauer and A.C. Wall, Quantum focusing conjecture, Phys. Rev. D 93 (2016) 064044 [arXiv:1506.02669] [INSPIRE].
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].
R.C. Myers, R. Pourhasan and M. Smolkin, On Spacetime Entanglement, JHEP 06 (2013) 013 [arXiv:1304.2030] [INSPIRE].
N. Engelhardt and A.C. Wall, Quantum Extremal Surfaces: Holographic Entanglement Entropy beyond the Classical Regime, JHEP 01 (2015) 073 [arXiv:1408.3203] [INSPIRE].
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].
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].
H. Casini, Relative entropy and the Bekenstein bound, Class. Quant. Grav. 25 (2008) 205021 [arXiv:0804.2182] [INSPIRE].
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].
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].
R.M. Wald, Black hole entropy is the Noether charge, Phys. Rev. D 48 (1993) R3427 [gr-qc/9307038] [INSPIRE].
T. Jacobson, G. Kang and R.C. Myers, On black hole entropy, Phys. Rev. D 49 (1994) 6587 [gr-qc/9312023] [INSPIRE].
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].
T. Faulkner, A. Lewkowycz and J. Maldacena, Quantum corrections to holographic entanglement entropy, JHEP 11 (2013) 074 [arXiv:1307.2892] [INSPIRE].
P. Calabrese and J. Cardy, Entanglement entropy and conformal field theory, J. Phys. A 42 (2009) 504005 [arXiv:0905.4013] [INSPIRE].
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].
D.V. Fursaev, A. Patrushev and S.N. Solodukhin, Distributional Geometry of Squashed Cones, Phys. Rev. D 88 (2013) 044054 [arXiv:1306.4000] [INSPIRE].
X. Dong, Holographic Entanglement Entropy for General Higher Derivative Gravity, JHEP 01 (2014) 044 [arXiv:1310.5713] [INSPIRE].
J. Camps, Generalized entropy and higher derivative Gravity, JHEP 03 (2014) 070 [arXiv:1310.6659] [INSPIRE].
J.H. Cooperman and M.A. Luty, Renormalization of Entanglement Entropy and the Gravitational Effective Action, JHEP 12 (2014) 045 [arXiv:1302.1878] [INSPIRE].
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].
R.M. Wald, Quantum field theory in curved spacetime and black hole thermodynamics, University of Chicago Press (1994).
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].
B.C. van Rees, Holographic renormalization for irrelevant operators and multi-trace counterterms, JHEP 08 (2011) 093 [arXiv:1102.2239] [INSPIRE].
B.C. van Rees, Irrelevant deformations and the holographic Callan-Symanzik equation, JHEP 10 (2011) 067 [arXiv:1105.5396] [INSPIRE].
V. Rosenhaus and M. Smolkin, Entanglement entropy, planar surfaces and spectral functions, JHEP 09 (2014) 119 [arXiv:1407.2891] [INSPIRE].
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].
M. Taylor and W. Woodhead, Renormalized entanglement entropy, JHEP 08 (2016) 165 [arXiv:1604.06808] [INSPIRE].
A. Lewkowycz and J. Maldacena, Generalized gravitational entropy, JHEP 08 (2013) 090 [arXiv:1304.4926] [INSPIRE].
T. Andrade and D. Marolf, AdS/CFT beyond the unitarity bound, JHEP 01 (2012) 049 [arXiv:1105.6337] [INSPIRE].
S. Minwalla, Restrictions imposed by superconformal invariance on quantum field theories, Adv. Theor. Math. Phys. 2 (1998) 781 [hep-th/9712074] [INSPIRE].
L.-Y. Hung, R.C. Myers and M. Smolkin, Some Calculable Contributions to Holographic Entanglement Entropy, JHEP 08 (2011) 039 [arXiv:1105.6055] [INSPIRE].
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].
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].
D. Marolf and S.F. Ross, Boundary Conditions and New Dualities: Vector Fields in AdS/CFT, JHEP 11 (2006) 085 [hep-th/0606113] [INSPIRE].
P. Minces and V.O. Rivelles, Energy and the AdS/CFT correspondence, JHEP 12 (2001) 010 [hep-th/0110189] [INSPIRE].
W. Mueck, An improved correspondence formula for AdS/CFT with multitrace operators, Phys. Lett. B 531 (2002) 301 [hep-th/0201100] [INSPIRE].
P. Minces, Multitrace operators and the generalized AdS/CFT prescription, Phys. Rev. D 68 (2003) 024027 [hep-th/0201172] [INSPIRE].
A. Sever and A. Shomer, A note on multitrace deformations and AdS/CFT, JHEP 07 (2002) 027 [hep-th/0203168] [INSPIRE].
S. Hollands, A. Ishibashi and D. Marolf, Counter-term charges generate bulk symmetries, Phys. Rev. D 72 (2005) 104025 [hep-th/0503105] [INSPIRE].
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].
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].
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].
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].
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].
S.N. Solodukhin, Nonminimal coupling and quantum entropy of black hole, Phys. Rev. D 56 (1997) 4968 [hep-th/9612061] [INSPIRE].
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].
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].
T. Nishioka, Relevant Perturbation of Entanglement Entropy and Stationarity, Phys. Rev. D 90 (2014) 045006 [arXiv:1405.3650] [INSPIRE].
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].
C.P. Herzog, Universal Thermal Corrections to Entanglement Entropy for Conformal Field Theories on Spheres, JHEP 10 (2014) 28 [arXiv:1407.1358] [INSPIRE].
J.S. Dowker, Expansion of Rényi entropy for free scalar fields, arXiv:1408.4055 [INSPIRE].
H. Casini, F.D. Mazzitelli and E. Testé, Area terms in entanglement entropy, Phys. Rev. D 91 (2015) 104035 [arXiv:1412.6522] [INSPIRE].
V. Rosenhaus and M. Smolkin, Entanglement Entropy for Relevant and Geometric Perturbations, JHEP 02 (2015) 015 [arXiv:1410.6530] [INSPIRE].
D.V. Fursaev, Energy, Hamiltonian, Noether charge and black holes, Phys. Rev. D 59 (1999) 064020 [hep-th/9809049] [INSPIRE].
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1607.01246
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Marolf, D., Wall, A.C. State-dependent divergences in the entanglement entropy. J. High Energ. Phys. 2016, 109 (2016). https://doi.org/10.1007/JHEP10(2016)109
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP10(2016)109