Abstract
We study quantum corrections to holographic mutual information for two disjoint spheres at a large separation by using the operator product expansion of the twist field. In the large separation limit, the holographic mutual information is vanishing at the semiclassical order, but receive quantum corrections from the fluctuations. We show that the leading contributions from the quantum fluctuations take universal forms as suggested from the boundary CFT. We find the universal behavior for the scalar, the vector, the tensor and the fermionic fields by treating these fields as free fields propagating in the fixed background and by using the 1/n prescription. In particular, for the fields with gauge symmetries, including the massless vector boson and massless graviton, we find that the gauge parts in the propagators play an indispensable role in reading the leading order corrections to the bulk mutual information.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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].
P. Calabrese and J.L. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech. 06 (2004) P06002 [hep-th/0405152] [INSPIRE].
P. Calabrese and J. Cardy, Entanglement entropy and quantum field theory: a non-technical introduction, Int. J. Quant. Inf. 4 (2006) 429 [quant-ph/050519] [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].
J.D. Bekenstein, Black holes and entropy, Phys. Rev. D 7 (1973) 2333 [INSPIRE].
J.M. Bardeen, B. Carter and S.W. Hawking, The four laws of black hole mechanics, Commun. Math. Phys. 31 (1973) 161 [INSPIRE].
A. Lewkowycz and J. Maldacena, Generalized gravitational entropy, JHEP 08 (2013) 090 [arXiv:1304.4926] [INSPIRE].
J.M. Maldacena, The large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].
S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [INSPIRE].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].
T. Nishioka, S. Ryu and T. Takayanagi, Holographic entanglement entropy: an overview, J. Phys. A 42 (2009) 504008 [arXiv:0905.0932] [INSPIRE].
M. Rangamani and T. Takayanagi, Holographic entanglement entropy, Lect. Notes Phys. 931 (2017) pp.1 [arXiv:1609.01287] [INSPIRE].
M.M. Wolf, F. Verstraete, M.B. Hastings and J.I. Cirac, Area laws in quantum systems: mutual information and correlations, Phys. Rev. Lett. 100 (2008) 070502 [arXiv:0704.3906].
M. Headrick, Entanglement Rényi entropies in holographic theories, Phys. Rev. D 82 (2010) 126010 [arXiv:1006.0047] [INSPIRE].
T. Faulkner, A. Lewkowycz and J. Maldacena, Quantum corrections to holographic entanglement entropy, JHEP 11 (2013) 074 [arXiv:1307.2892] [INSPIRE].
T. Barrella, X. Dong, S.A. Hartnoll and V.L. Martin, Holographic entanglement beyond classical gravity, JHEP 09 (2013) 109 [arXiv:1306.4682] [INSPIRE].
B. Chen and J.-J. Zhang, On short interval expansion of Rényi entropy, JHEP 11 (2013) 164 [arXiv:1309.5453] [INSPIRE].
B. Chen, J. Long and J.-J. Zhang, Holographic Rényi entropy for CFT with W symmetry, JHEP 04 (2014) 041 [arXiv:1312.5510] [INSPIRE].
E. Perlmutter, Comments on Rényi entropy in AdS 3 /CFT 2, JHEP 05 (2014) 052 [arXiv:1312.5740] [INSPIRE].
B. Chen, F.-Y. Song and J.-J. Zhang, Holographic Rényi entropy in AdS 3 /LCFT 2 correspondence, JHEP 03 (2014) 137 [arXiv:1401.0261] [INSPIRE].
M. Beccaria and G. Macorini, On the next-to-leading holographic entanglement entropy in AdS 3 /CFT 2, JHEP 04 (2014) 045 [arXiv:1402.0659] [INSPIRE].
M. Headrick, A. Maloney, E. Perlmutter and I.G. Zadeh, Rényi entropies, the analytic bootstrap and 3D quantum gravity at higher genus, JHEP 07 (2015) 059 [arXiv:1503.07111] [INSPIRE].
B. Chen and J.-Q. Wu, 1-loop partition function in AdS 3 /CFT 2, JHEP 12 (2015) 109 [arXiv:1509.02062] [INSPIRE].
J.-J. Zhang, Holographic Rényi entropy for two-dimensional N = (1, 1) superconformal field theory, JHEP 12 (2015) 027 [arXiv:1510.01423] [INSPIRE].
Z. Li and J.-J. Zhang, On one-loop entanglement entropy of two short intervals from OPE of twist operators, JHEP 05 (2016) 130 [arXiv:1604.02779] [INSPIRE].
Z. Li and J.-J. Zhang, Holographic Rényi entropy for two-dimensional N = (2, 2) superconformal field theory, Phys. Rev. D 95 (2017) 126009 [arXiv:1611.00546] [INSPIRE].
P. Calabrese, J. Cardy and E. Tonni, Entanglement entropy of two disjoint intervals in conformal field theory II, J. Stat. Mech. 01 (2011) P01021 [arXiv:1011.5482] [INSPIRE].
J. Cardy, Some results on the mutual information of disjoint regions in higher dimensions, J. Phys. A 46 (2013) 285402 [arXiv:1304.7985] [INSPIRE].
L.-Y. Hung, R.C. Myers and M. Smolkin, Twist operators in higher dimensions, JHEP 10 (2014) 178 [arXiv:1407.6429] [INSPIRE].
N. Shiba, Entanglement entropy of two spheres, JHEP 07 (2012) 100 [arXiv:1201.4865] [INSPIRE].
H. Casini and M. Huerta, Remarks on the entanglement entropy for disconnected regions, JHEP 03 (2009) 048 [arXiv:0812.1773] [INSPIRE].
C.A. Agón, I. Cohen-Abbo and H.J. Schnitzer, Large distance expansion of mutual information for disjoint disks in a free scalar theory, JHEP 11 (2016) 073 [arXiv:1505.03757] [INSPIRE].
B. Chen and J. Long, Rényi mutual information for a free scalar field in even dimensions, Phys. Rev. D 96 (2017) 045006 [arXiv:1612.00114] [INSPIRE].
J. Long, On co-dimension two defect operators, arXiv:1611.02485 [INSPIRE].
C. Agón and T. Faulkner, Quantum corrections to holographic mutual information, JHEP 08 (2016) 118 [arXiv:1511.07462] [INSPIRE].
B. Chen, L. Chen, P.-X. Hao and J. Long, On the mutual information in conformal field theory, JHEP 06 (2017) 096 [arXiv:1704.03692] [INSPIRE].
H. Casini, M. Huerta and R.C. Myers, Towards a derivation of holographic entanglement entropy, JHEP 05 (2011) 036 [arXiv:1102.0440] [INSPIRE].
X. Dong, The gravity dual of Rényi entropy, Nature Commun. 7 (2016) 12472 [arXiv:1601.06788] [INSPIRE].
D.E. Berenstein, R. Corrado, W. Fischler and J.M. Maldacena, The operator product expansion for Wilson loops and surfaces in the large N limit, Phys. Rev. D 59 (1999) 105023 [hep-th/9809188] [INSPIRE].
J. Gomis and T. Okuda, S-duality, ’t Hooft operators and the operator product expansion, JHEP 09 (2009) 072 [arXiv:0906.3011] [INSPIRE].
B. Chen, C.-Y. Liu and J.-B. Wu, Operator product expansion of Wilson surfaces from M5-branes, JHEP 01 (2008) 007 [arXiv:0711.2194] [INSPIRE].
F.A. Dolan and H. Osborn, Conformal four point functions and the operator product expansion, Nucl. Phys. B 599 (2001) 459 [hep-th/0011040] [INSPIRE].
F.A. Dolan and H. Osborn, Conformal partial waves and the operator product expansion, Nucl. Phys. B 678 (2004) 491 [hep-th/0309180] [INSPIRE].
M. Hogervorst, H. Osborn and S. Rychkov, Diagonal limit for conformal blocks in d dimensions, JHEP 08 (2013) 014 [arXiv:1305.1321] [INSPIRE].
B. Chen and J.-Q. Wu, Large interval limit of Rényi entropy at high temperature, Phys. Rev. D 92 (2015) 126002 [arXiv:1412.0763] [INSPIRE].
B. Chen and J.-Q. Wu, Universal relation between thermal entropy and entanglement entropy in conformal field theories, Phys. Rev. D 91 (2015) 086012 [arXiv:1412.0761] [INSPIRE].
B. Chen and J.-Q. Wu, Holographic calculation for large interval Rényi entropy at high temperature, Phys. Rev. D 92 (2015) 106001 [arXiv:1506.03206] [INSPIRE].
K. Krasnov, Holography and Riemann surfaces, Adv. Theor. Math. Phys. 4 (2000) 929 [hep-th/0005106] [INSPIRE].
T. Hartman, Entanglement entropy at large central charge, arXiv:1303.6955 [INSPIRE].
T. Faulkner, The entanglement Rényi entropies of disjoint intervals in AdS/CFT, arXiv:1303.7221 [INSPIRE].
I.I. Kogan, S. Mouslopoulos and A. Papazoglou, The m → 0 limit for massive graviton in dS 4 and AdS 4 : how to circumvent the van Dam-Veltman-Zakharov discontinuity, Phys. Lett. B 503 (2001) 173 [hep-th/0011138] [INSPIRE].
M. Porrati, No van Dam-Veltman-Zakharov discontinuity in AdS space, Phys. Lett. B 498 (2001) 92 [hep-th/0011152] [INSPIRE].
E. D’Hoker, D.Z. Freedman, S.D. Mathur, A. Matusis and L. Rastelli, Graviton and gauge boson propagators in AdS d+1, Nucl. Phys. B 562 (1999) 330 [hep-th/9902042] [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].
W. Donnelly, Entanglement entropy and non-Abelian gauge symmetry, Class. Quant. Grav. 31 (2014) 214003 [arXiv:1406.7304] [INSPIRE].
W. Donnelly and L. Freidel, Local subsystems in gauge theory and gravity, JHEP 09 (2016) 102 [arXiv:1601.04744] [INSPIRE].
E. D’Hoker, D.Z. Freedman, S.D. Mathur, A. Matusis and L. Rastelli, Graviton and gauge boson propagators in AdS d+1, Nucl. Phys. B 562 (1999) 330 [hep-th/9902042] [INSPIRE].
E. D’Hoker, D.Z. Freedman, S.D. Mathur, A. Matusis and L. Rastelli, Graviton exchange and complete four point functions in the AdS/CFT correspondence, Nucl. Phys. B 562 (1999) 353 [hep-th/9903196] [INSPIRE].
M. Henningson and K. Sfetsos, Spinors and the AdS/CFT correspondence, Phys. Lett. B 431 (1998) 63 [hep-th/9803251] [INSPIRE].
T. Kawano and K. Okuyama, Spinor exchange in AdS d+1, Nucl. Phys. B 565 (2000) 427 [hep-th/9905130] [INSPIRE].
C.P. Herzog and M. Spillane, Thermal corrections to Rényi entropies for free fermions, JHEP 04 (2016) 124 [arXiv:1506.06757] [INSPIRE].
W. Mück and K.S. Viswanathan, Conformal field theory correlators from classical field theory on anti-de Sitter space II. Vector and spinor fields, Phys. Rev. D 58 (1998) 106006 [hep-th/9805145] [INSPIRE].
A. Basu and L.I. Uruchurtu, Gravitino propagator in anti de Sitter space, Class. Quant. Grav. 23 (2006) 6059 [hep-th/0603089] [INSPIRE].
N. Iqbal and H. Liu, Real-time response in AdS/CFT with application to spinors, Fortsch. Phys. 57 (2009) 367 [arXiv:0903.2596] [INSPIRE].
N. Drukker, J. Gomis and S. Matsuura, Probing N = 4 SYM with surface operators, JHEP 10 (2008) 048 [arXiv:0805.4199] [INSPIRE].
A. Naqvi, Propagators for massive symmetric tensor and p-forms in AdS d+1, JHEP 12 (1999) 025 [hep-th/9911182] [INSPIRE].
M.S. Costa, V. Gonçalves and J. Penedones, Spinning AdS propagators, JHEP 09 (2014) 064 [arXiv:1404.5625] [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: 1712.05131
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, 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 licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Chen, B., Fan, ZY., Li, WM. et al. Holographic mutual information of two disjoint spheres. J. High Energ. Phys. 2018, 113 (2018). https://doi.org/10.1007/JHEP04(2018)113
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP04(2018)113