Abstract
We first present an analysis of infinitesimal null deformations for the entanglement entropy, which leads to a major simplification of the proof of the C, F and A-theorems in quantum field theory. Next, we study the quantum null energy condition (QNEC) on the light-cone for a CFT. Finally, we combine these tools in order to establish the irreversibility of renormalization group flows on planar d-dimensional defects, embedded in D-dimensional conformal field theories. This proof completes and unifies all known defect irreversibility theorems for defect dimensions d ≤ 4. The F-theorem on defects (d = 3) is a new result using information-theoretic methods. For d ≥ 4 we also establish the monotonicity of the relative entropy coefficient proportional to Rd−4. The geometric construction connects the proof of irreversibility with and without defects through the QNEC inequality in the bulk, and makes contact with the proof of strong subadditivity of holographic entropy taking into account quantum corrections.
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References
I. Affleck and A.W.W. Ludwig, Universal noninteger ‘ground state degeneracy’ in critical quantum systems, Phys. Rev. Lett. 67 (1991) 161 [INSPIRE].
D. Friedan and A. Konechny, On the boundary entropy of one-dimensional quantum systems at low temperature, Phys. Rev. Lett. 93 (2004) 030402 [hep-th/0312197] [INSPIRE].
A.B. Zamolodchikov, Irreversibility of the Flux of the Renormalization Group in a 2D Field Theory, JETP Lett. 43 (1986) 730 [INSPIRE].
J.L. Cardy, Is There a c Theorem in Four-Dimensions?, Phys. Lett. B 215 (1988) 749 [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].
S.N. Solodukhin, Entanglement entropy, conformal invariance and extrinsic geometry, Phys. Lett. B 665 (2008) 305 [arXiv:0802.3117] [INSPIRE].
H. Casini, M. Huerta and R.C. Myers, Towards a derivation of holographic entanglement entropy, JHEP 05 (2011) 036 [arXiv:1102.0440] [INSPIRE].
R.C. Myers and A. Sinha, Seeing a c-theorem with holography, Phys. Rev. D 82 (2010) 046006 [arXiv:1006.1263] [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].
P. Calabrese and J. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech. 2004 (2004) P06002.
K. Jensen, A. O’Bannon, B. Robinson and R. Rodgers, From the Weyl Anomaly to Entropy of Two-Dimensional Boundaries and Defects, Phys. Rev. Lett. 122 (2019) 241602 [arXiv:1812.08745] [INSPIRE].
N. Kobayashi, T. Nishioka, Y. Sato and K. Watanabe, Towards a C-theorem in defect CFT, JHEP 2019 (2019) 39 [arXiv:1810.06995].
Z. Komargodski and A. Schwimmer, On Renormalization Group Flows in Four Dimensions, JHEP 12 (2011) 099 [arXiv:1107.3987] [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].
T. Shachar, R. Sinha and M. Smolkin, RG flows on two-dimensional spherical defects, arXiv:2212.08081 [INSPIRE].
Y. Wang, Defect a-theorem and a-maximization, JHEP 02 (2022) 061 [arXiv:2101.12648] [INSPIRE].
G. Cuomo, Z. Komargodski and A. Raviv-Moshe, Renormalization Group Flows on Line Defects, Phys. Rev. Lett. 128 (2022) 021603 [arXiv:2108.01117] [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, On the RG running of the entanglement entropy of a circle, Phys. Rev. D 85 (2012) 125016 [arXiv:1202.5650] [INSPIRE].
H. Casini, E. Testé and G. Torroba, Markov Property of the Conformal Field Theory Vacuum and the a Theorem, Phys. Rev. Lett. 118 (2017) 261602 [arXiv:1704.01870] [INSPIRE].
H. Casini, I. Salazar Landea and G. Torroba, The g-theorem and quantum information theory, JHEP 10 (2016) 140 [arXiv:1607.00390] [INSPIRE].
H. Casini, I. Salazar Landea and G. Torroba, Entropic g Theorem in General Spacetime Dimensions, Phys. Rev. Lett. 130 (2023) 111603 [arXiv:2212.10575] [INSPIRE].
H. Casini, E. Testé and G. Torroba, Relative entropy and the RG flow, JHEP 03 (2017) 089 [arXiv:1611.00016] [INSPIRE].
H. Casini, I. Salazar Landea and G. Torroba, Irreversibility in quantum field theories with boundaries, JHEP 04 (2019) 166 [arXiv:1812.08183] [INSPIRE].
D. Gaiotto, Boundary F-maximization, arXiv:1403.8052 [INSPIRE].
R. Bousso, Z. Fisher, S. Leichenauer and A.C. Wall, Quantum focusing conjecture, Phys. Rev. D 93 (2016) 064044 [arXiv:1506.02669] [INSPIRE].
J. Koeller and S. Leichenauer, Holographic Proof of the Quantum Null Energy Condition, Phys. Rev. D 94 (2016) 024026 [arXiv:1512.06109] [INSPIRE].
H. Casini, E. Testé and G. Torroba, Modular Hamiltonians on the null plane and the Markov property of the vacuum state, J. Phys. A 50 (2017) 364001 [arXiv:1703.10656] [INSPIRE].
H. Casini, E. Testé and G. Torroba, All the entropies on the light-cone, JHEP 05 (2018) 005 [arXiv:1802.04278] [INSPIRE].
M. Mezei, Entanglement entropy across a deformed sphere, Phys. Rev. D 91 (2015) 045038 [arXiv:1411.7011] [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].
D. Petz, Quantum Information Theory and Quantum Statistics, Springer (2008) [https://doi.org/10.1007/978-3-540-74636-2].
R. Bousso et al., Proof of the Quantum Null Energy Condition, Phys. Rev. D 93 (2016) 024017 [arXiv:1509.02542] [INSPIRE].
S. Balakrishnan, T. Faulkner, Z.U. Khandker and H. Wang, A General Proof of the Quantum Null Energy Condition, JHEP 09 (2019) 020 [arXiv:1706.09432] [INSPIRE].
S. Balakrishnan et al., Entropy variations and light ray operators from replica defects, JHEP 09 (2022) 217 [arXiv:1906.08274] [INSPIRE].
S. Leichenauer, A. Levine and A. Shahbazi-Moghaddam, Energy density from second shape variations of the von Neumann entropy, Phys. Rev. D 98 (2018) 086013 [arXiv:1802.02584] [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 [Erratum ibid. 87 (2013) 069904] [arXiv:1105.3445] [INSPIRE].
R. Longo, Entropy distribution of localised states, Commun. Math. Phys. 373 (2019) 473 [arXiv:1809.03358] [INSPIRE].
M. Billò, V. Gonçalves, E. Lauria and M. Meineri, Defects in conformal field theory, JHEP 04 (2016) 091 [arXiv:1601.02883] [INSPIRE].
M. Mezei and J. Virrueta, The Quantum Null Energy Condition and Entanglement Entropy in Quenches, arXiv:1909.00919 [INSPIRE].
H. Casini, F.D. Mazzitelli and E. Testé, Area terms in entanglement entropy, Phys. Rev. D 91 (2015) 104035 [arXiv:1412.6522] [INSPIRE].
L. Daguerre, M. Ginzburg and G. Torroba, Holographic entanglement entropy inequalities beyond strong subadditivity, JHEP 10 (2022) 199 [arXiv:2208.03334] [INSPIRE].
T. Azeyanagi, A. Karch, T. Takayanagi and E.G. Thompson, Holographic calculation of boundary entropy, JHEP 03 (2008) 054 [arXiv:0712.1850] [INSPIRE].
M.-K. Yuan and Y. Zhou, Defect Localized Entropy: Renormalization Group and Holography, arXiv:2209.08835 [INSPIRE].
R.C. Myers and A. Sinha, Holographic c-theorems in arbitrary dimensions, JHEP 01 (2011) 125 [arXiv:1011.5819] [INSPIRE].
A.C. Wall, Maximin Surfaces, and the Strong Subadditivity of the Covariant Holographic Entanglement Entropy, Class. Quant. Grav. 31 (2014) 225007 [arXiv:1211.3494] [INSPIRE].
O. Aharony, M. Berkooz and E. Silverstein, Multiple trace operators and nonlocal string theories, JHEP 08 (2001) 006 [hep-th/0105309] [INSPIRE].
E. Witten, Multitrace operators, boundary conditions, and AdS/CFT correspondence, hep-th/0112258 [INSPIRE].
T. Faulkner, A. Lewkowycz and J. Maldacena, Quantum corrections to holographic entanglement entropy, JHEP 11 (2013) 074 [arXiv:1307.2892] [INSPIRE].
C. Akers, N. Engelhardt, G. Penington and M. Usatyuk, Quantum Maximin Surfaces, JHEP 08 (2020) 140 [arXiv:1912.02799] [INSPIRE].
V. Benedetti, H. Casini and P.J. Martinez, Mutual information of generalized free fields, Phys. Rev. D 107 (2023) 046003 [arXiv:2210.00013] [INSPIRE].
Acknowledgments
We thank discussions with Eduardo Testé around the calculations presented in section 2. HC and GT are supported by CONICET (PIP grant 11220200101008CO), ANPCyT (PICT 2018–2517), CNEA, and Instituto Balseiro, Universidad Nacional de Cuyo. HC acknowledges an “It From Qubit” grant of the Simons Foundation. ISL would like to thank IB and ICTP for hospitality.
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Casini, H., Landea, I.S. & Torroba, G. Irreversibility, QNEC, and defects. J. High Energ. Phys. 2023, 4 (2023). https://doi.org/10.1007/JHEP07(2023)004
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DOI: https://doi.org/10.1007/JHEP07(2023)004