Abstract
We study the dynamics of the geometric entanglement entropy of a 2D CFT in the presence of a boundary. We show that this dynamics is governed by local equations of motion, that take the same form as 2D Jackiw-Teitelboim gravity coupled to the CFT. If we assume that the boundary has a small thickness ϵ and constant boundary entropy, we derive that its location satisfies the equations of motion of Schwarzian quantum mechanics with coupling constant C = c ϵ/12π. We rederive this result via energy-momentum conservation.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
T. Jacobson, Thermodynamics of space-time: the Einstein equation of state, Phys. Rev. Lett. 75 (1995) 1260 [gr-qc/9504004] [INSPIRE].
R. Bousso, Z. Fisher, S. Leichenauer and A.C. Wall, Quantum focusing conjecture, Phys. Rev. D 93 (2016) 064044 [arXiv:1506.02669] [INSPIRE].
S. Balakrishnan, T. Faulkner, Z.U. Khandker and H. Wang, A general proof of the quantum null energy condition, arXiv:1706.09432 [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].
J. Koeller, S. Leichenauer, A. Levine and A. Shahbazi-Moghaddam, Local modular hamiltonians from the quantum null energy condition, Phys. Rev. D 97 (2018) 065011 [arXiv:1702.00412] [INSPIRE].
R. Jackiw, Lower dimensional gravity, Nucl. Phys. B 252 (1985) 343 [INSPIRE].
C. Teitelboim, Gravitation and hamiltonian structure in two space-time dimensions, Phys. Lett. B 126 (1983) 41.
J. Engelsöy, T.G. Mertens and H. Verlinde, An investigation of AdS 2 backreaction and holography, JHEP 07 (2016) 139 [arXiv:1606.03438] [INSPIRE].
A. Kitaev, Hidden correlations in the Hawking radiation and thermal noise, talk given at the Fundamental Physics Prize Symposium, November 10, Stanford University, Stanford, U.S.A. (2014).
A. Kitaev, Hidden correlations in the Hawking radiation and thermal noise, talk given at KITP, February 12, Santa Barbara, U.S.A. (2015).
A. Kitaev, A simple model of quantum holography, talks given at KITP, April 7 and May 27, Santa Barbara, U.S.A. (2015).
J. Maldacena and D. Stanford, Remarks on the Sachdev-Ye-Kitaev model, Phys. Rev. D 94 (2016) 106002 [arXiv:1604.07818] [INSPIRE].
K. Jensen, Chaos in AdS 2 holography, Phys. Rev. Lett. 117 (2016) 111601 [arXiv:1605.06098] [INSPIRE].
J. Maldacena, D. Stanford and Z. Yang, Conformal symmetry and its breaking in two dimensional nearly Anti-de-Sitter space, PTEP 2016 (2016) 12C104 [arXiv:1606.01857] [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].
J. Cardy and E. Tonni, Entanglement hamiltonians in two-dimensional conformal field theory, J. Stat. Mech. 1612 (2016) 123103 [arXiv:1608.01283] [INSPIRE].
J. de Boer, F.M. Haehl, M.P. Heller and R.C. Myers, Entanglement, holography and causal diamonds, JHEP 08 (2016) 162 [arXiv:1606.03307] [INSPIRE].
H. Casini, M. Huerta and R.C. Myers, Towards a derivation of holographic entanglement entropy, JHEP 05 (2011) 036 [arXiv:1102.0440] [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. D 87 (2013) 069904] [arXiv:1105.3445] [INSPIRE].
A.C. Wall, Testing the generalized second law in 1 + 1 dimensional conformal vacua: an argument for the causal horizon, Phys. Rev. D 85 (2012) 024015 [arXiv:1105.3520] [INSPIRE].
A.C. Wall, A survey of black hole thermodynamics, arXiv:1804.10610 [INSPIRE].
Z.U. Khandker, S. Kundu and D. Li, Bulk matter and the boundary quantum null energy condition, JHEP 08 (2018) 162 [arXiv:1803.03997] [INSPIRE].
A. Almheiri and J. Polchinski, Models of AdS 2 backreaction and holography, JHEP 11 (2015) 014 [arXiv:1402.6334] [INSPIRE].
T.D. Chung and H.L. Verlinde, Dynamical moving mirrors and black holes, Nucl. Phys. B 418 (1994) 305 [hep-th/9311007] [INSPIRE].
R. Bott, On the characteristic classes of groups of diffeomorphisms, Enseign. Math. 23 (1977) 209.
T.G. Mertens, G.J. Turiaci and H.L. Verlinde, Solving the Schwarzian via the Conformal Bootstrap, JHEP 08 (2017) 136 [arXiv:1705.08408] [INSPIRE].
G. Turiaci and H. Verlinde, On CFT and quantum chaos, JHEP 12 (2016) 110 [arXiv:1603.03020] [INSPIRE].
J. Haegeman, T.J. Osborne, H. Verschelde and F. Verstraete, Entanglement renormalization for quantum fields in real space, Phys. Rev. Lett. 110 (2013) 100402 [arXiv:1102.5524] [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
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
Callebaut, N., Verlinde, H. Entanglement dynamics in 2D CFT with boundary: entropic origin of JT gravity and Schwarzian QM. J. High Energ. Phys. 2019, 45 (2019). https://doi.org/10.1007/JHEP05(2019)045
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP05(2019)045