Abstract
We study the effect of inhomogeneity, which is induced by the graviton mass in massive gravity, on the mutual information and the chaotic behavior of a 2+1-dimensional field theory from the gauge/gravity duality. When the system is near-homogeneous, the mutual information increases as the graviton mass grows. However, when the system is far from homogeneity, the mutual information decreases as the graviton mass increases. By adding the perturbations of energy into the system, we investigate the dynamical mutual information in the shock wave geometry. We find that the greater perturbations disrupt the mutual information more rapidly, which resembles the butterfly effect in chaos theory. Besides, the greater inhomogeneity reduces the dynamical mutual information more quickly just as in the static case.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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].
S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [INSPIRE].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].
S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].
M. Van Raamsdonk, Lectures on gravity and entanglement, arXiv:1609.00026 [INSPIRE].
M. Rangamani and T. Takayanagi, Holographic entanglement entropy, Lect. Notes Phys. 931 (2017) 1 [arXiv:1609.01287] [INSPIRE].
V. Balasubramanian et al., Thermalization of strongly coupled field theories, Phys. Rev. Lett. 106 (2011) 191601 [arXiv:1012.4753] [INSPIRE].
R.-G. Cai, S. He, L. Li and Y.-L. Zhang, Holographic entanglement entropy in insulator/superconductor transition, JHEP 07 (2012) 088 [arXiv:1203.6620] [INSPIRE].
R.-G. Cai, S. He, L. Li and Y.-L. Zhang, Holographic entanglement entropy on P-wave superconductor phase transition, JHEP 07 (2012) 027 [arXiv:1204.5962] [INSPIRE].
C.V. Johnson, Large N phase transitions, finite volume and entanglement entropy, JHEP 03 (2014) 047 [arXiv:1306.4955] [INSPIRE].
X. Bai, B.-H. Lee, L. Li, J.-R. Sun and H.-Q. Zhang, Time evolution of entanglement entropy in quenched holographic superconductors, JHEP 04 (2015) 066 [arXiv:1412.5500] [INSPIRE].
Y. Ling, P. Liu, C. Niu, J.-P. Wu and Z.-Y. Xian, Holographic entanglement entropy close to quantum phase transitions, JHEP 04 (2016) 114 [arXiv:1502.03661] [INSPIRE].
X.-X. Zeng and L.-F. Li, Van der Waals phase transition in the framework of holography, Phys. Lett. B 764 (2017) 100 [arXiv:1512.08855] [INSPIRE].
X. Zeng and W. Liu, Holographic thermalization in Gauss-Bonnet gravity, Phys. Lett. B 726 (2013) 481 [arXiv:1305.4841] [INSPIRE].
X.-X. Zeng, X.-M. Liu and W.-B. Liu, Holographic thermalization with a chemical potential in Gauss-Bonnet gravity, JHEP 03 (2014) 031 [arXiv:1311.0718] [INSPIRE].
X.-X. Zeng, X.-M. Liu and W.-B. Liu, Holographic thermalization in noncommutative geometry, Phys. Lett. B 744 (2015) 48 [arXiv:1407.5262] [INSPIRE].
X.-X. Zeng, X.-M. Liu and L.-F. Li, Phase structure of the Born-Infeld-anti-de Sitter black holes probed by non-local observables, Eur. Phys. J. C 76 (2016) 616 [arXiv:1601.01160] [INSPIRE].
M.A. Nielsen and I.L. Chuang, Quantum computation and quantum information, 10th Anniversary Edition, Cambridge University Press (2010).
J.M. Maldacena, Eternal black holes in anti-de Sitter, JHEP 04 (2003) 021 [hep-th/0106112] [INSPIRE].
J. Maldacena and L. Susskind, Cool horizons for entangled black holes, Fortschr. Phys. 61 (2013) 781 [arXiv:1306.0533] [INSPIRE].
M. Van Raamsdonk, Comments on quantum gravity and entanglement, arXiv:0907.2939 [INSPIRE].
I.A. Morrison and M.M. Roberts, Mutual information between thermo-field doubles and disconnected holographic boundaries, JHEP 07 (2013) 081 [arXiv:1211.2887] [INSPIRE].
S.H. Shenker and D. Stanford, Black holes and the butterfly effect, JHEP 03 (2014) 067 [arXiv:1306.0622] [INSPIRE].
S. Leichenauer, Disrupting entanglement of black holes, Phys. Rev. D 90 (2014) 046009 [arXiv:1405.7365] [INSPIRE].
D. Berenstein and A.M. Garcia-Garcia, Universal quantum constraints on the butterfly effect, arXiv:1510.08870 [INSPIRE].
N. Sircar, J. Sonnenschein and W. Tangarife, Extending the scope of holographic mutual information and chaotic behavior, JHEP 05 (2016) 091 [arXiv:1602.07307] [INSPIRE].
Y. Ling, P. Liu and J.-P. Wu, Holographic butterfly effect at quantum critical points, arXiv:1610.02669 [INSPIRE].
M.M. Qaemmaqami, Criticality in third order Lovelock gravity and butterfly effect, arXiv:1705.05235 [INSPIRE].
M. Alishahiha, A. Davody, A. Naseh and S.F. Taghavi, On butterfly effect in higher derivative gravities, JHEP 11 (2016) 032 [arXiv:1610.02890] [INSPIRE].
C. de Rham and G. Gabadadze, Generalization of the Fierz-Pauli action, Phys. Rev. D 82 (2010) 044020 [arXiv:1007.0443] [INSPIRE].
C. de Rham, G. Gabadadze and A.J. Tolley, Resummation of massive gravity, Phys. Rev. Lett. 106 (2011) 231101 [arXiv:1011.1232] [INSPIRE].
D. Vegh, Holography without translational symmetry, arXiv:1301.0537 [INSPIRE].
R.A. Davison, Momentum relaxation in holographic massive gravity, Phys. Rev. D 88 (2013) 086003 [arXiv:1306.5792] [INSPIRE].
M. Blake, D. Tong and D. Vegh, Holographic lattices give the graviton an effective mass, Phys. Rev. Lett. 112 (2014) 071602 [arXiv:1310.3832] [INSPIRE].
R.-G. Cai, Y.-P. Hu, Q.-Y. Pan and Y.-L. Zhang, Thermodynamics of black holes in massive gravity, Phys. Rev. D 91 (2015) 024032 [arXiv:1409.2369] [INSPIRE].
Y.-P. Hu and H. Zhang, Misner-sharp mass and the unified first law in massive gravity, Phys. Rev. D 92 (2015) 024006 [arXiv:1502.00069] [INSPIRE].
R.-G. Cai and R.-Q. Yang, Insulator/metal phase transition and colossal magnetoresistance in holographic model, Phys. Rev. D 92 (2015) 106002 [arXiv:1507.03105] [INSPIRE].
L.-M. Cao and Y. Peng, Counterterms in massive gravity theory, Phys. Rev. D 92 (2015) 124052 [arXiv:1509.08738] [INSPIRE].
Y.-P. Hu, H.-F. Li, H.-B. Zeng and H.-Q. Zhang, Holographic Josephson junction from massive gravity, Phys. Rev. D 93 (2016) 104009 [arXiv:1512.07035] [INSPIRE].
Y.-P. Hu, X.-X. Zeng and H.-Q. Zhang, Holographic thermalization and generalized Vaidya-AdS solutions in massive gravity, Phys. Lett. B 765 (2017) 120 [arXiv:1611.00677] [INSPIRE].
X.-X. Zeng, H. Zhang and L.-F. Li, Phase transition of holographic entanglement entropy in massive gravity, Phys. Lett. B 756 (2016) 170 [arXiv:1511.00383] [INSPIRE].
Y.-P. Hu, F. Pan and X.-M. Wu, The effects of massive graviton on the equilibrium between the black hole and radiation gas in an isolated box, arXiv:1703.08599 [INSPIRE].
R.-G. Cai, L. Li, L.-F. Li and R.-Q. Yang, Introduction to holographic superconductor models, Sci. China Phys. Mech. Astron. 58 (2015) 060401 [arXiv:1502.00437] [INSPIRE].
T. Dray and G. ’t Hooft, The gravitational shock wave of a massless particle, Nucl. Phys. B 253 (1985) 173 [INSPIRE].
W.H. Huang and Y.H. Du, Butterfly effect and holographic mutual information under external field and spatial noncommutativity, JHEP 02 (2017) 032 [arXiv:1609.08841] [INSPIRE].
S. Hellerman, Lattice gauge theories have gravitational duals, hep-th/0207226 [INSPIRE].
T. Andrade and B. Withers, A simple holographic model of momentum relaxation, JHEP 05 (2014) 101 [arXiv:1311.5157] [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: 1704.03989
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, 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 license, and indicate if changes were made.
About this article
Cite this article
Cai, RG., Zeng, XX. & Zhang, HQ. Influence of inhomogeneities on holographic mutual information and butterfly effect. J. High Energ. Phys. 2017, 82 (2017). https://doi.org/10.1007/JHEP07(2017)082
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP07(2017)082