Abstract
In this paper, we consider the evolution of the thermofield-double state under the double-traced operator that connects its both sides. We will compute the entanglement entropy of the resulting state using the replica trick for the large N field theory. To leading order, it can be computed from the two-point function of the theory, where, in CFTs, it is fixed by the symmetries. Due to the exponential decay of the interaction, the entanglement entropy saturates about the thermal time after the interaction is on. Next, we restrict ourselves to one dimension and assume that the theory at strong coupling is effectively described by the Schwarzian action. We then compute the coarse-grained entropy of the resulting state using the four-point function. The equality of the two entropies implies that the double-traced operators in our theory act coherently. In AdS/CFT correspondence where the thermofield-double state corresponds to a two-sided black hole, the action of a double-traced operator corresponds to shrinking or expanding the black hole in the bulk.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
R.M. Wald, General relativity, Chicago University Press, Chicago U.S.A. (1984).
S.W. Hawking and G.F.R. Ellis, The large scale structure of space-time, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge U.K. (2011) [INSPIRE].
I. Racz and R.M. Wald, Extensions of spacetimes with killing horizons, Class. Quant. Grav. 9 (1992) 2643.
I. Racz and R.M. Wald, Global extensions of space-times describing asymptotic final states of black holes, Class. Quant. Grav. 13 (1996) 539 [gr-qc/9507055] [INSPIRE].
W. Israel, Event horizons in static vacuum space-times, Phys. Rev. 164 (1967) 1776 [INSPIRE].
W. Israel, Event horizons in static electrovac space-times, Commun. Math. Phys. 8 (1968) 245.
B. Carter, Axisymmetric black hole has only two degrees of freedom, Phys. Rev. Lett. 26 (1971) 331 [INSPIRE].
J.M. Bardeen, B. Carter and S.W. Hawking, The four laws of black hole mechanics, Commun. Math. Phys. 31 (1973) 161.
J.M. Maldacena, The large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [Adv. Theor. Math. Phys. 2 (1998) 231] [hep-th/9711200] [INSPIRE].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].
J.M. Maldacena, Eternal black holes in anti-de Sitter, JHEP 04 (2003) 021 [hep-th/0106112] [INSPIRE].
W. Israel, Thermo-field dynamics of black holes, Phys. Lett. A 57 (1976) 107.
P. Gao, D.L. Jafferis and A.C. Wall, Traversable wormholes via a double trace deformation, JHEP 12 (2017) 151 [arXiv:1608.05687] [INSPIRE].
J. Maldacena, D. Stanford and Z. Yang, Diving into traversable wormholes, Fortsch. Phys. 65 (2017) 1700034 [arXiv:1704.05333] [INSPIRE].
J. Maldacena and X.-L. Qi, Eternal traversable wormhole, arXiv:1804.00491 [INSPIRE].
Y. Chen and P. Zhang, Entanglement entropy of two coupled SYK models and eternal traversable wormhole, JHEP 07 (2019) 033 [arXiv:1903.10532] [INSPIRE].
C. Teitelboim, Gravitation and Hamiltonian structure in two space-time dimensions, Phys. Lett. B 126 (1983) 41 [INSPIRE].
R. Jackiw, Lower dimensional gravity, Nucl. Phys. B 252 (1985) 343 [INSPIRE].
A. Kitaev and S.J. Suh, The soft mode in the Sachdev-Ye-Kitaev model and its gravity dual, JHEP 05 (2018) 183 [arXiv:1711.08467] [INSPIRE].
J. Maldacena and D. Stanford, Remarks on the Sachdev-Ye-Kitaev model, Phys. Rev. D 94 (2016) 106002 [arXiv:1604.07818] [INSPIRE].
S. Sachdev and J. Ye, Gapless spin fluid ground state in a random, quantum Heisenberg magnet, Phys. Rev. Lett. 70 (1993) 3339 [cond-mat/9212030] [INSPIRE].
A. Kitaev and S.J. Suh, Statistical mechanics of a two-dimensional black hole, JHEP 05 (2019) 198 [arXiv:1808.07032] [INSPIRE].
A. Almheiri and J. Polchinski, Models of AdS2 backreaction and holography, JHEP 11 (2015) 014 [arXiv:1402.6334] [INSPIRE].
A. Fabbri and J. Navarro-Salas, Modeling black hole evaporation, World Scientific, Singapore (2005).
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].
J.J. Bisognano and E.H. Wichmann, On the duality condition for quantum fields, J. Math. Phys. 17 (1976) 303 [INSPIRE].
E. Witten, APS medal for exceptional achievement in research: invited article on entanglement properties of quantum field theory, Rev. Mod. Phys. 90 (2018) 045003 [arXiv:1803.04993] [INSPIRE].
R. Haag, Local quantum physics: Fields, particles, algebras, Springer, Germany (1992).
S.H. Shenker and D. Stanford, Stringy effects in scrambling, JHEP 05 (2015) 132 [arXiv:1412.6087] [INSPIRE].
J. Maldacena and L. Susskind, Cool horizons for entangled black holes, Fortsch. Phys. 61 (2013) 781 [arXiv:1306.0533] [INSPIRE].
T. Dray and G. ’t Hooft, The gravitational shock wave of a massless particle, Nucl. Phys. B 253 (1985) 173.
D. Bagrets, A. Altland and A. Kamenev, Power-law out of time order correlation functions in the SYK model, Nucl. Phys. B 921 (2017) 727 [arXiv:1702.08902] [INSPIRE].
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].
H.T. Lam, T.G. Mertens, G.J. Turiaci and H. Verlinde, Shockwave S-matrix from Schwarzian quantum mechanics, JHEP 11 (2018) 182 [arXiv:1804.09834] [INSPIRE].
Z. Yang, The quantum gravity dynamics of near extremal black holes, JHEP 05 (2019) 205 [arXiv:1809.08647] [INSPIRE].
J. Maldacena, S.H. Shenker and D. Stanford, A bound on chaos, JHEP 08 (2016) 106 [arXiv:1503.01409] [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 1905.02305
Rights and permissions
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.
About this article
Cite this article
Dadras, P. Disentangling the thermofield-double state. J. High Energ. Phys. 2022, 75 (2022). https://doi.org/10.1007/JHEP01(2022)075
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP01(2022)075