Abstract
We perform a systematic study of the maximum Lyapunov exponent values λ for the motion of classical closed strings in Anti-de Sitter black hole geometries with spherical, planar and hyperbolic horizons. Analytical estimates from the linearized varia- tional equations together with numerical integrations predict the bulk Lyapunov exponent value as λ ≈ 2πTn, where n is the winding number of the string. The celebrated bound on chaos stating that λ ≤ 2πT is thus systematically modified for winding strings in the bulk. Within gauge/string duality, such strings apparently correspond to complicated operators which either do not move on Regge trajectories, or move on subleading trajectories with an unusual slope. Depending on the energy scale, the out-of-time-ordered correlation functions of these operators may still obey the bound 2πT, or they may violate it like the bulk exponent. We do not know exactly why the bound on chaos can be modified but the indication from the gauge/string dual viewpoint is that the correlation functions of the dual gauge operators never factorize and thus the original derivation of the bound on chaos does not apply.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J. Maldacena, S.H. Shenker and D. Stanford, A bound on chaos, JHEP08 (2016) 106 [arXiv:1503.01409] [INSPIRE].
Y. Sekino and L. Susskind, Fast scramblers, JHEP10 (2008) 065 [arXiv:0808.2096] [INSPIRE].
N. Lashkari, D. Stanford, M. Hastings, T. Osborne and P. Hayden, Towards the fast scrambling conjecture, JHEP04 (2013) 022 [arXiv:1111.6580] [INSPIRE].
S.H. Shenker and D. Stanford, Black holes and the butterfly effect, JHEP03 (2014) 067 [arXiv:1306.0622] [INSPIRE].
D.A. Roberts, D. Stanford and L. Susskind, Localized shocks, JHEP03 (2015) 051 [arXiv:1409.8180] [INSPIRE].
S.H. Shenker and D. Stanford, Stringy effects in scrambling, JHEP05 (2015) 132 [arXiv:1412.6087] [INSPIRE].
J. Polchinski, Black hole S matrix, arXiv:1505.08108 [INSPIRE].
T. Scaffidi and E. Altman, Chaos in a classical limit of the Sachdev-Ye-Kitaev model, Phys. Rev.B 100 (2019) 155128 [arXiv:1711.04768] [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].
O. Parcolet and A. Georges, Non-Fermi liquid regime of a doped Mott insulator, Phys. Rev.B 59 (1998) 5341 [cond-mat/9806119].
A. Kitaev, A simple model of quantum holography (part 1), talk at KITP, April 7, 2015, http://online.kitp.ucsb.edu/online/entangled15/kitaev/.
A. Kitaev, A simple model of quantum holography (part 2), talk at KITP, May 27, 2015, http://online.kitp.ucsb.edu/online/entangled15/kitaev2/.
E. Marcus and S. Vandoren, A new class of SYK-like models with maximal chaos, JHEP01 (2019) 166 [arXiv:1808.01190] [INSPIRE].
A.L. Fitzpatrick and J. Kaplan, A quantum correction to chaos, JHEP05 (2016) 070 [arXiv:1601.06164] [INSPIRE].
K. Hashimoto and N. Tanahashi, Universality in chaos of particle motion near black hole horizon, Phys. Rev.D 95 (2017) 024007 [arXiv:1610.06070] [INSPIRE].
S. Dalui, B.R. Majhi and P. Mishra, Presence of horizon makes particle motion chaotic, Phys. Lett.B 788 (2019) 486 [arXiv:1803.06527] [INSPIRE].
J. de Boer, E. Llabrés, J.F. Pedraza and D. Vegh, Chaotic strings in AdS/CFT, Phys. Rev. Lett.120 (2018) 201604 [arXiv:1709.01052] [INSPIRE].
J.R. David, S. Khetrapal and S.P. Kumar, Local quenches and quantum chaos from perturbations, JHEP10 (2017) 156 [arXiv:1707.07166] [INSPIRE].
P. Basu and L.A. Pando Zayas, Analytic nonintegrability in string theory, Phys. Rev.D 84 (2011) 046006 [arXiv:1105.2540] [INSPIRE].
A. Stepanchuk and A.A. Tseytlin, On (non)integrability of classical strings in p-brane backgrounds, J. Phys.A 46 (2013) 125401 [arXiv:1211.3727] [INSPIRE].
Y. Chervonyi and O. Lunin, (Non)-integrability of geodesics in D-brane backgrounds, JHEP02 (2014) 061 [arXiv:1311.1521] [INSPIRE].
C. Núñez, J.M. Penín, D. Roychowdhury and J. Van Gorsel, The non-integrability of strings in massive type IIA and their holographic duals, JHEP06 (2018) 078 [arXiv:1802.04269] [INSPIRE].
D. Giataganas, L.A. Pando Zayas and K. Zoubos, On marginal deformations and non-integrability, JHEP01 (2014) 129 [arXiv:1311.3241] [INSPIRE].
N. Beisert et al., Review of AdS/CFT Integrability: An Overview, Lett. Math. Phys.99 (2012) 3 [arXiv:1012.3982] [INSPIRE].
A.V. Frolov and A.L. Larsen, Chaotic scattering and capture of strings by a black hole, Class. Quant. Grav.16 (1999) 3717 [gr-qc/9908039] [INSPIRE].
L.A. Pando Zayas and C.A. Terrero-Escalante, Chaos in the gauge/gravity correspondence, JHEP09 (2010) 094 [arXiv:1007.0277] [INSPIRE].
P. Basu and L.A. Pando Zayas, Chaos rules out integrability of strings on AdS 5× T 1,1, Phys. Lett.B 700 (2011) 243 [arXiv:1103.4107] [INSPIRE].
P. Basu, D. Das and A. Ghosh, Integrability lost: Chaotic dynamics of classical strings on a confining holographic background, Phys. Lett.B 699 (2011) 388 [arXiv:1103.4101] [INSPIRE].
P. Basu, P. Chaturvedi and P. Samantray, Chaotic dynamics of strings in charged black hole backgrounds, Phys. Rev.D 95 (2017) 066014 [arXiv:1607.04466] [INSPIRE].
Y. Asano, D. Kawai, H. Kyono and K. Yoshida, Chaotic strings in a near Penrose limit of AdS 5× T 1,1 , JHEP08 (2015) 060 [arXiv:1505.07583] [INSPIRE].
Y. Asano, H. Kyono and K. Yoshida, Melnikov’s method in string theory, JHEP09 (2016) 103 [arXiv:1607.07302] [INSPIRE].
D. Giataganas and K. Sfetsos, Non-integrability in non-relativistic theories, JHEP06 (2014) 018 [arXiv:1403.2703] [INSPIRE].
T. Ishii, K. Murata and K. Yoshida, Fate of chaotic strings in a confining geometry, Phys. Rev.D 95 (2017) 066019 [arXiv:1610.05833] [INSPIRE].
R.B. Mann, Pair production of topological anti-de Sitter black holes, Class. Quant. Grav.14 (1997) L109 [gr-qc/9607071] [INSPIRE].
D.R. Brill, J. Louko and P. Peldan, Thermodynamics of (3+1)-dimensional black holes with toroidal or higher genus horizons, Phys. Rev.D 56 (1997) 3600 [gr-qc/9705012] [INSPIRE].
L. Vanzo, Black holes with unusual topology, Phys. Rev.D 56 (1997) 6475 [gr-qc/9705004] [INSPIRE].
D. Birmingham, Topological black holes in anti-de Sitter space, Class. Quant. Grav.16 (1999) 1197 [hep-th/9808032] [INSPIRE].
R.B. Mann, Topological black holes — outside looking in, Annals Israel Phys. Soc.13 (1997) 311 [gr-qc/9709039] [INSPIRE].
W.L. Smith and R.B. Mann, Formation of topological black holes from gravitational collapse, Phys. Rev.D 56 (1997) 4942 [gr-qc/9703007] [INSPIRE].
Y.C. Ong, Hawking evaporation time scale of topological black Holes in anti-de Sitter spacetime, Nucl. Phys.B 903 (2016) 387 [arXiv:1507.07845] [INSPIRE].
Y. Chen and E. Teo, Black holes with bottle-shaped horizons, Phys. Rev.D 93 (2016) 124028 [arXiv:1604.07527] [INSPIRE].
C.V. Johnson and F. Rosso, Holographic heat engines, entanglement entropy and renormalization group flow, Class. Quant. Grav.36 (2019) 015019 [arXiv:1806.05170] [INSPIRE].
R. Emparan, AdS/CFT duals of topological black holes and the entropy of zero-energy states, JHEP06 (1999) 036 [hep-th/9906040] [INSPIRE].
A.L. Larsen, Chaotic string-capture by black hole, Class. Quant. Grav.11 (1994) 1201 [hep-th/9309086] [INSPIRE].
A.E. Motter, Relativistic chaos is coordinate invariant, Phys. Rev. Lett.91 (2003) 231101 [gr-qc/0305020] [INSPIRE].
N.L. Balazs and A. Voros, Chaos on the pseudosphere, Phys. Rept.143 (1986) 109 [INSPIRE].
S.S. Gubser, I.R. Klebanov and A.M. Polyakov, A semi-classical limit of the gauge/string correspondence, Nucl. Phys.B 636 (2002) 99 [hep-th/0204051] [INSPIRE].
H.J. de Vega and I.L. Egusquiza, Planetoid string solutions in 3 + 1 axissymmetric spaces, Phys. Rev.D 54 (1996) 7513 [hep-th/9607056] [INSPIRE].
R.C. Brower, J. Polchinski, M.J. Strassler and C.-I. Tan, The pomeron and gauge/string duality, JHEP12 (2007) 005 [hep-th/0603115] [INSPIRE].
D. Giataganas and K. Zoubos, Non-integrability and chaos with unquenched flavor, JHEP10 (2017) 042 [arXiv:1707.04033] [INSPIRE].
E. Perlmutter, Bounding the space of holographic CFTs with chaos, JHEP10 (2016) 069 [arXiv:1602.08272] [INSPIRE].
P. Basu and K. Jaswin, Higher point OTOCs and the bound on chaos, arXiv:1809.05331 [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: 1904.06295
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
Čubrović, M. The bound on chaos for closed strings in Anti-de Sitter black hole backgrounds. J. High Energ. Phys. 2019, 150 (2019). https://doi.org/10.1007/JHEP12(2019)150
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP12(2019)150