Abstract
We study the holographic entanglement entropy of spatial regions with corners in the AdS4/BCFT3 correspondence by considering three dimensional boundary conformal field theories whose boundary is a timelike plane. We compute analytically the corner function corresponding to an infinite wedge having one edge on the boundary. A relation between this corner function and the holographic one point function of the stress tensor is observed. An analytic expression for the corner function of an infinite wedge having only its tip on the boundary is also provided. This formula requires to find the global minimum among two extrema of the area functional. The corresponding critical configurations of corners are studied. The results have been checked against a numerical analysis performed by computing the area of the minimal surfaces anchored to some finite domains containing corners.
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L. Amico, R. Fazio, A. Osterloh and V. Vedral, Entanglement in many-body systems, Rev. Mod. Phys. 80 (2008) 517 [quant-ph/0703044] [INSPIRE].
J. Eisert, M. Cramer and M.B. Plenio, Area laws for the entanglement entropy — a review, Rev. Mod. Phys. 82 (2010) 277 [arXiv:0808.3773] [INSPIRE].
P. Calabrese, J. Cardy and B. Doyon eds., Entanglement entropy in extended quantum systems, J. Phys. A 42 (2009) 500301.
M. Rangamani and T. Takayanagi, Holographic entanglement entropy, Lect. Notes Phys. 931 (2017) pp.1-246 [arXiv:1609.01287] [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].
M. Freedman and M. Headrick, Bit threads and holographic entanglement, Commun. Math. Phys. 352 (2017) 407 [arXiv:1604.00354] [INSPIRE].
C.G. Callan Jr. and F. Wilczek, On geometric entropy, Phys. Lett. B 333 (1994) 55 [hep-th/9401072] [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].
G. Vidal, J.I. Latorre, E. Rico and A. Kitaev, Entanglement in quantum critical phenomena, Phys. Rev. Lett. 90 (2003) 227902 [quant-ph/0211074] [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, J. Cardy and E. Tonni, Entanglement entropy of two disjoint intervals in conformal field theory, J. Stat. Mech. 11 (2009) P11001 [arXiv:0905.2069] [INSPIRE].
M. Headrick, Entanglement Rényi entropies in holographic theories, Phys. Rev. D 82 (2010) 126010 [arXiv:1006.0047] [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].
A. Coser, L. Tagliacozzo and E. Tonni, On Rényi entropies of disjoint intervals in conformal field theory, J. Stat. Mech. 01 (2014) P01008 [arXiv:1309.2189] [INSPIRE].
R.D. Sorkin, 1983 paper on entanglement entropy: “on the entropy of the vacuum outside a horizon”, in Tenth international conference on general relativity and gravitation (held Padova Italy, 4-9 July 1983), contributed papers, vol. II, (1983), pg. 734 [arXiv:1402.3589] [INSPIRE].
L. Bombelli, R.K. Koul, J. Lee and R.D. Sorkin, A quantum source of entropy for black holes, Phys. Rev. D 34 (1986) 373 [INSPIRE].
M. Srednicki, Entropy and area, Phys. Rev. Lett. 71 (1993) 666 [hep-th/9303048] [INSPIRE].
R.C. Myers and A. Sinha, Seeing a c-theorem with holography, Phys. Rev. D 82 (2010) 046006 [arXiv:1006.1263] [INSPIRE].
R.C. Myers and A. Sinha, Holographic c-theorems in arbitrary dimensions, JHEP 01 (2011) 125 [arXiv:1011.5819] [INSPIRE].
D.L. Jafferis, I.R. Klebanov, S.S. Pufu and B.R. Safdi, Towards the F-theorem: N = 2 field theories on the three-sphere, JHEP 06 (2011) 102 [arXiv:1103.1181] [INSPIRE].
I.R. Klebanov, S.S. Pufu and B.R. Safdi, F-theorem without supersymmetry, JHEP 10 (2011) 038 [arXiv:1105.4598] [INSPIRE].
H. Liu and M. Mezei, A refinement of entanglement entropy and the number of degrees of freedom, JHEP 04 (2013) 162 [arXiv:1202.2070] [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].
J.M. Maldacena, Wilson loops in large-N field theories, Phys. Rev. Lett. 80 (1998) 4859 [hep-th/9803002] [INSPIRE].
N. Drukker, D.J. Gross and H. Ooguri, Wilson loops and minimal surfaces, Phys. Rev. D 60 (1999) 125006 [hep-th/9904191] [INSPIRE].
E. Fradkin and J.E. Moore, Entanglement entropy of 2D conformal quantum critical points: hearing the shape of a quantum drum, Phys. Rev. Lett. 97 (2006) 050404 [cond-mat/0605683] [INSPIRE].
H. Casini and M. Huerta, Universal terms for the entanglement entropy in 2 + 1 dimensions, Nucl. Phys. B 764 (2007) 183 [hep-th/0606256] [INSPIRE].
T. Hirata and T. Takayanagi, AdS/CFT and strong subadditivity of entanglement entropy, JHEP 02 (2007) 042 [hep-th/0608213] [INSPIRE].
H. Casini, M. Huerta and L. Leitao, Entanglement entropy for a Dirac fermion in three dimensions: vertex contribution, Nucl. Phys. B 814 (2009) 594 [arXiv:0811.1968] [INSPIRE].
S. Whitsitt, W. Witczak-Krempa and S. Sachdev, Entanglement entropy of the large-N Wilson-Fisher conformal field theory, Phys. Rev. B 95 (2017) 045148 [arXiv:1610.06568] [INSPIRE].
E.H. Lieb and M.B. Ruskai, Proof of the strong subadditivity of quantum-mechanical entropy, J. Math. Phys. 14 (1973) 1938 [INSPIRE].
A. Kallin, K. Hyatt, R. Singh and R. Melko, Entanglement at a two-dimensional quantum critical point: a numerical linked-cluster expansion study, Phys. Rev. Lett. 110 (2013) 135702 [arXiv:1212.5269].
E.M. Stoudenmire, P. Gustainis, R. Johal, S. Wessel and R.G. Melko, Corner contribution to the entanglement entropy of strongly interacting O(2) quantum critical systems in 2 + 1 dimensions, Phys. Rev. B 90 (2014) 235106 [arXiv:1409.6327] [INSPIRE].
A.B. Kallin, E.M. Stoudenmire, P. Fendley, R.R.P. Singh and R.G. Melko, Corner contribution to the entanglement entropy of an O(3) quantum critical point in 2 + 1 dimensions, J. Stat. Mech. 06 (2014) P06009 [arXiv:1401.3504] [INSPIRE].
J.-M. Stéphan, S. Furukawa, G. Misguich and V. Pasquier, Shannon and entanglement entropies of one- and two-dimensional critical wave functions, Phys. Rev. B 80 (2009) 184421 [arXiv:0906.1153].
T. Devakul and R. Singh, Quantum critical universality and singular corner entanglement entropy of bilayer Heisenberg-Ising model, Phys. Rev. B 90 (2014) 064424 [arXiv:1406.0185].
J. Helmes, L.E. Hayward Sierens, A. Chandran, W. Witczak-Krempa and R.G. Melko, Universal corner entanglement of Dirac fermions and gapless bosons from the continuum to the lattice, Phys. Rev. B 94 (2016) 125142 [arXiv:1606.03096] [INSPIRE].
P. Bueno, R.C. Myers and W. Witczak-Krempa, Universality of corner entanglement in conformal field theories, Phys. Rev. Lett. 115 (2015) 021602 [arXiv:1505.04804] [INSPIRE].
P. Bueno and R.C. Myers, Corner contributions to holographic entanglement entropy, JHEP 08 (2015) 068 [arXiv:1505.07842] [INSPIRE].
T. Faulkner, R.G. Leigh and O. Parrikar, Shape dependence of entanglement entropy in conformal field theories, JHEP 04 (2016) 088 [arXiv:1511.05179] [INSPIRE].
J.L. Cardy, Conformal invariance and surface critical behavior, Nucl. Phys. B 240 (1984) 514 [INSPIRE].
J.L. Cardy, Boundary conditions, fusion rules and the Verlinde formula, Nucl. Phys. B 324 (1989) 581 [INSPIRE].
P. Di Francesco, P. Mathieu and D. Senechal, Conformal field theory, Springer-Verlag, New York U.S.A., (1997) [INSPIRE].
J.L. Cardy, Boundary conformal field theory, hep-th/0411189 [INSPIRE].
D.M. McAvity and H. Osborn, Energy momentum tensor in conformal field theories near a boundary, Nucl. Phys. B 406 (1993) 655 [hep-th/9302068] [INSPIRE].
D.M. McAvity and H. Osborn, Conformal field theories near a boundary in general dimensions, Nucl. Phys. B 455 (1995) 522 [cond-mat/9505127] [INSPIRE].
K. Jensen and A. O’Bannon, Constraint on defect and boundary renormalization group flows, Phys. Rev. Lett. 116 (2016) 091601 [arXiv:1509.02160] [INSPIRE].
S.N. Solodukhin, Boundary terms of conformal anomaly, Phys. Lett. B 752 (2016) 131 [arXiv:1510.04566] [INSPIRE].
C.P. Herzog, K.-W. Huang and K. Jensen, Universal entanglement and boundary geometry in conformal field theory, JHEP 01 (2016) 162 [arXiv:1510.00021] [INSPIRE].
D. Fursaev, Conformal anomalies of CFT’s with boundaries, JHEP 12 (2015) 112 [arXiv:1510.01427] [INSPIRE].
K.-W. Huang, Boundary anomalies and correlation functions, JHEP 08 (2016) 013 [arXiv:1604.02138] [INSPIRE].
C.P. Herzog and K.-W. Huang, Boundary conformal field theory and a boundary central charge, JHEP 10 (2017) 189 [arXiv:1707.06224] [INSPIRE].
K. Jensen and A. O’Bannon, Holography, entanglement entropy and conformal field theories with boundaries or defects, Phys. Rev. D 88 (2013) 106006 [arXiv:1309.4523] [INSPIRE].
D.V. Fursaev and S.N. Solodukhin, Anomalies, entropy and boundaries, Phys. Rev. D 93 (2016) 084021 [arXiv:1601.06418] [INSPIRE].
C. Berthiere and S.N. Solodukhin, Boundary effects in entanglement entropy, Nucl. Phys. B 910 (2016) 823 [arXiv:1604.07571] [INSPIRE].
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].
R.C. Myers and A. Singh, Entanglement entropy for singular surfaces, JHEP 09 (2012) 013 [arXiv:1206.5225] [INSPIRE].
P. Fonda, L. Giomi, A. Salvio and E. Tonni, On shape dependence of holographic mutual information in AdS 4, JHEP 02 (2015) 005 [arXiv:1411.3608] [INSPIRE].
P. Fonda, D. Seminara and E. Tonni, On shape dependence of holographic entanglement entropy in AdS 4/CFT 3, JHEP 12 (2015) 037 [arXiv:1510.03664] [INSPIRE].
M.R. Mohammadi Mozaffar, A. Mollabashi and F. Omidi, Holographic mutual information for singular surfaces, JHEP 12 (2015) 082 [arXiv:1511.00244] [INSPIRE].
T. Takayanagi, Holographic dual of BCFT, Phys. Rev. Lett. 107 (2011) 101602 [arXiv:1105.5165] [INSPIRE].
M. Fujita, T. Takayanagi and E. Tonni, Aspects of AdS/BCFT, JHEP 11 (2011) 043 [arXiv:1108.5152] [INSPIRE].
M. Nozaki, T. Takayanagi and T. Ugajin, Central charges for BCFTs and holography, JHEP 06 (2012) 066 [arXiv:1205.1573] [INSPIRE].
C. Bachas, Asymptotic symmetries of AdS 2-branes, hep-th/0205115 [INSPIRE].
A. Karch and L. Randall, Locally localized gravity, JHEP 05 (2001) 008 [hep-th/0011156] [INSPIRE].
O. DeWolfe, D.Z. Freedman and H. Ooguri, Holography and defect conformal field theories, Phys. Rev. D 66 (2002) 025009 [hep-th/0111135] [INSPIRE].
T. Azeyanagi, A. Karch, T. Takayanagi and E.G. Thompson, Holographic calculation of boundary entropy, JHEP 03 (2008) 054 [arXiv:0712.1850] [INSPIRE].
J. Erdmenger, C. Hoyos, A. O’Bannon and J. Wu, A holographic model of the Kondo effect, JHEP 12 (2013) 086 [arXiv:1310.3271] [INSPIRE].
J. Erdmenger, M. Flory, C. Hoyos, M.-N. Newrzella and J.M.S. Wu, Entanglement entropy in a holographic Kondo model, Fortsch. Phys. 64 (2016) 109 [arXiv:1511.03666] [INSPIRE].
R.-X. Miao, C.-S. Chu and W.-Z. Guo, New proposal for a holographic boundary conformal field theory, Phys. Rev. D 96 (2017) 046005 [arXiv:1701.04275] [INSPIRE].
C.-S. Chu, R.-X. Miao and W.-Z. Guo, On new proposal for holographic BCFT, JHEP 04 (2017) 089 [arXiv:1701.07202] [INSPIRE].
A. Faraji Astaneh and S.N. Solodukhin, Holographic calculation of boundary terms in conformal anomaly, Phys. Lett. B 769 (2017) 25 [arXiv:1702.00566] [INSPIRE].
A. Faraji Astaneh, C. Berthiere, D. Fursaev and S.N. Solodukhin, Holographic calculation of entanglement entropy in the presence of boundaries, Phys. Rev. D 95 (2017) 106013 [arXiv:1703.04186] [INSPIRE].
K. Brakke, The surface evolver, Experiment. Math. 1 (1992) 141.
Surface Evolver program webpage, http://www.susqu.edu/brakke/evolver/evolver.html.
G. Hayward, Gravitational action for space-times with nonsmooth boundaries, Phys. Rev. D 47 (1993) 3275 [INSPIRE].
S.W. Hawking and C.J. Hunter, The gravitational Hamiltonian in the presence of nonorthogonal boundaries, Class. Quant. Grav. 13 (1996) 2735 [gr-qc/9603050] [INSPIRE].
M. Henningson and K. Skenderis, The holographic Weyl anomaly, JHEP 07 (1998) 023 [hep-th/9806087] [INSPIRE].
M. Bianchi, D.Z. Freedman and K. Skenderis, How to go with an RG flow, JHEP 08 (2001) 041 [hep-th/0105276] [INSPIRE].
M. Bianchi, D.Z. Freedman and K. Skenderis, Holographic renormalization, Nucl. Phys. B 631 (2002) 159 [hep-th/0112119] [INSPIRE].
K. Skenderis, Lecture notes on holographic renormalization, Class. Quant. Grav. 19 (2002) 5849 [hep-th/0209067] [INSPIRE].
V. Balasubramanian and P. Kraus, A stress tensor for anti-de Sitter gravity, Commun. Math. Phys. 208 (1999) 413 [hep-th/9902121] [INSPIRE].
R.C. Myers, Stress tensors and Casimir energies in the AdS/CFT correspondence, Phys. Rev. D 60 (1999) 046002 [hep-th/9903203] [INSPIRE].
S. de Haro, S.N. Solodukhin and K. Skenderis, Holographic reconstruction of space-time and renormalization in the AdS/CFT correspondence, Commun. Math. Phys. 217 (2001) 595 [hep-th/0002230] [INSPIRE].
H. Casini, M. Huerta and R.C. Myers, Towards a derivation of holographic entanglement entropy, JHEP 05 (2011) 036 [arXiv:1102.0440] [INSPIRE].
J.K. Erickson, G.W. Semenoff and K. Zarembo, Wilson loops in N = 4 supersymmetric Yang-Mills theory, Nucl. Phys. B 582 (2000) 155 [hep-th/0003055] [INSPIRE].
K. Nagasaki, H. Tanida and S. Yamaguchi, Holographic interface-particle potential, JHEP 01 (2012) 139 [arXiv:1109.1927] [INSPIRE].
E. Tonni, Holographic entanglement entropy: near horizon geometry and disconnected regions, JHEP 05 (2011) 004 [arXiv:1011.0166] [INSPIRE].
E. Tonni, Corner contributions to holographic entanglement entropy in AdS 4/BCFT 3, talk delivered at Integrability in Gauge and String Theory (IGST 2017), École Normale Supérieure, Paris France, 20 July 2017.
M. Preti, D. Trancanelli and E. Vescovi, Quark-antiquark potential in defect conformal field theory, JHEP 10 (2017) 079 [arXiv:1708.04884] [INSPIRE].
N. Drukker and V. Forini, Generalized quark-antiquark potential at weak and strong coupling, JHEP 06 (2011) 131 [arXiv:1105.5144] [INSPIRE].
D. Correa, J. Maldacena and A. Sever, The quark anti-quark potential and the cusp anomalous dimension from a TBA equation, JHEP 08 (2012) 134 [arXiv:1203.1913] [INSPIRE].
H. Liu and A.A. Tseytlin, D = 4 super Yang-Mills, D = 5 gauged supergravity and D = 4 conformal supergravity, Nucl. Phys. B 533 (1998) 88 [hep-th/9804083] [INSPIRE].
A. Buchel, J. Escobedo, R.C. Myers, M.F. Paulos, A. Sinha and M. Smolkin, Holographic GB gravity in arbitrary dimensions, JHEP 03 (2010) 111 [arXiv:0911.4257] [INSPIRE].
D. Deutsch and P. Candelas, Boundary effects in quantum field theory, Phys. Rev. D 20 (1979) 3063 [INSPIRE].
R.-X. Miao and C.-S. Chu, Universality for shape dependence of Casimir effects from Weyl anomaly, arXiv:1706.09652 [INSPIRE].
H. Bateman, Higher transcendental functions, volume I, McGraw-Hill, New York U.S.A., (1953).
M. Abramowitz and I. Stegun, Handbook of mathematical functions, Dover Publications, New York U.S.A., (1972).
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Seminara, D., Sisti, J. & Tonni, E. Corner contributions to holographic entanglement entropy in AdS4/BCFT3. J. High Energ. Phys. 2017, 76 (2017). https://doi.org/10.1007/JHEP11(2017)076
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DOI: https://doi.org/10.1007/JHEP11(2017)076