Abstract
A global quench is an interesting setting where we can study thermalization of subsystems in a pure state. We investigate entanglement entropy (EE) growth in global quenches in holographic field theories and relate some of its aspects to quantities characterizing chaos. More specifically we obtain four key results:
-
1.
We prove holographic bounds on the entanglement velocity v E and the butterfly effect speed v B that arises in the study of chaos.
-
2.
We obtain the EE as a function of time for large spherical entangling surfaces analytically. We show that the EE is insensitive to the details of the initial state or quench protocol.
-
3.
In a thermofield double state we determine analytically the two-sided mutual information between two large concentric spheres separated in time.
-
4.
We derive a bound on the rate of growth of EE for arbitrary shapes, and develop an expansion for EE at early times.
In a companion paper [1], these results are put in the broader context of EE growth in chaotic systems: we relate EE growth to the chaotic spreading of operators, derive bounds on EE at a given time, and compare the holographic results to spin chain numerics and toy models. In this paper, we perform holographic calculations that provide the basis of arguments presented in that paper.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
M. Mezei and D. Stanford, On entanglement spreading in chaotic systems, arXiv:1608.05101 [INSPIRE].
J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [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].
S. Ryu and T. Takayanagi, Aspects of Holographic Entanglement Entropy, JHEP 08 (2006) 045 [hep-th/0605073] [INSPIRE].
V.E. Hubeny, M. Rangamani and T. Takayanagi, A covariant holographic entanglement entropy proposal, JHEP 07 (2007) 062 [arXiv:0705.0016] [INSPIRE].
P. Calabrese and J.L. Cardy, Evolution of entanglement entropy in one-dimensional systems, J. Stat. Mech. 0504 (2005) P04010 [cond-mat/0503393] [INSPIRE].
C.T. Asplund, A. Bernamonti, F. Galli and T. Hartman, Entanglement Scrambling in 2d Conformal Field Theory, JHEP 09 (2015) 110 [arXiv:1506.03772] [INSPIRE].
J. Cardy, Quantum Quenches to a Critical Point in One Dimension: some further results, J. Stat. Mech. 1602 (2016) 023103 [arXiv:1507.07266] [INSPIRE].
G. De Chiara, S. Montangero, P. Calabrese and R. Fazio, Entanglement entropy dynamics in Heisenberg chains, J. Stat. Mech. 0603 (2006) P03001 [cond-mat/0512586] [INSPIRE].
P. Calabrese and J. Cardy, Quantum Quenches in Extended Systems, J. Stat. Mech. 0706 (2007) P06008 [arXiv:0704.1880] [INSPIRE].
M. Fagotti and P. Calabrese, Evolution of entanglement entropy following a quantum quench: Analytic results for the XY chain in a transverse magnetic field, Phys. Rev. A 78 (2008) 010306 [arXiv:0804.3559].
A.M. Läuchli and C. Kollath, Spreading of correlations and entanglement after a quench in the one-dimensional Bose Hubbard model, J. Stat. Mech. Theor. Exp. 5 (2008) 05018 [arXiv:0803.2947].
H. Kim and D.A. Huse, Ballistic spreading of entanglement in a diffusive nonintegrable system, Phys. Rev. Lett. 111 (2013) 127205 [arXiv:1306.4306].
Y. Lemonik and A. Mitra, Entanglement properties of the critical quench of O (N) bosons, Phys. Rev. B 94 (2016) 024306 [arXiv:1512.02749].
J.S. Cotler, M.P. Hertzberg, M. Mezei and M.T. Mueller, Entanglement Growth after a Global Quench in Free Scalar Field Theory, JHEP 11 (2016) 166 [arXiv:1609.00872] [INSPIRE].
T. Hartman and J. Maldacena, Time Evolution of Entanglement Entropy from Black Hole Interiors, JHEP 05 (2013) 014 [arXiv:1303.1080] [INSPIRE].
H. Liu and S.J. Suh, Entanglement Tsunami: Universal Scaling in Holographic Thermalization, Phys. Rev. Lett. 112 (2014) 011601 [arXiv:1305.7244] [INSPIRE].
H. Liu and S.J. Suh, Entanglement growth during thermalization in holographic systems, Phys. Rev. D 89 (2014) 066012 [arXiv:1311.1200] [INSPIRE].
J. Abajo-Arrastia, J. Aparicio and E. Lopez, Holographic Evolution of Entanglement Entropy, JHEP 11 (2010) 149 [arXiv:1006.4090] [INSPIRE].
T. Albash and C.V. Johnson, Evolution of Holographic Entanglement Entropy after Thermal and Electromagnetic Quenches, New J. Phys. 13 (2011) 045017 [arXiv:1008.3027] [INSPIRE].
V. Balasubramanian et al., Thermalization of Strongly Coupled Field Theories, Phys. Rev. Lett. 106 (2011) 191601 [arXiv:1012.4753] [INSPIRE].
V. Balasubramanian et al., Holographic Thermalization, Phys. Rev. D 84 (2011) 026010 [arXiv:1103.2683] [INSPIRE].
S.H. Shenker and D. Stanford, Black holes and the butterfly effect, JHEP 03 (2014) 067 [arXiv:1306.0622] [INSPIRE].
D.A. Roberts, D. Stanford and L. Susskind, Localized shocks, JHEP 03 (2015) 051 [arXiv:1409.8180] [INSPIRE].
J. Maldacena, S.H. Shenker and D. Stanford, A bound on chaos, JHEP 08 (2016) 106 [arXiv:1503.01409] [INSPIRE].
Y. Gu, X.-L. Qi and D. Stanford, Local criticality, diffusion and chaos in generalized Sachdev-Ye-Kitaev models, arXiv:1609.07832 [INSPIRE].
T. Hartman and N. Afkhami-Jeddi, Speed Limits for Entanglement, arXiv:1512.02695 [INSPIRE].
S. Kundu and J.F. Pedraza, Spread of entanglement for small subsystems in holographic CFTs, Phys. Rev. D 95 (2017) 086008 [arXiv:1602.05934] [INSPIRE].
H. Casini, H. Liu and M. Mezei, Spread of entanglement and causality, JHEP 07 (2016) 077 [arXiv:1509.05044] [INSPIRE].
A. Nahum, J. Ruhman, S. Vijay and J. Haah, Quantum Entanglement Growth Under Random Unitary Dynamics, arXiv:1608.06950 [INSPIRE].
J.M. Maldacena, Eternal black holes in anti-de Sitter, JHEP 04 (2003) 021 [hep-th/0106112] [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].
P. Hosur, X.-L. Qi, D.A. Roberts and B. Yoshida, Chaos in quantum channels, JHEP 02 (2016) 004 [arXiv:1511.04021] [INSPIRE].
S. Bravyi, M.B. Hastings and F. Verstraete, Lieb-Robinson Bounds and the Generation of Correlations and Topological Quantum Order, Phys. Rev. Lett. 97 (2006) 050401 [quant-ph/0603121].
M. Mariën, K.M.R. Audenaert, K. Van Acoleyen and F. Verstraete, Entanglement Rates and the Stability of the Area Law for the Entanglement Entropy, ArXiv e-prints (2014) [arXiv:1411.0680].
S.G. Avery and M.F. Paulos, Universal Bounds on the Time Evolution of Entanglement Entropy, Phys. Rev. Lett. 113 (2014) 231604 [arXiv:1407.0705] [INSPIRE].
H. Liu and M. Mezei, Probing renormalization group flows using entanglement entropy, JHEP 01 (2014) 098 [arXiv:1309.6935] [INSPIRE].
V.E. Hubeny and H. Maxfield, Holographic probes of collapsing black holes, JHEP 03 (2014) 097 [arXiv:1312.6887] [INSPIRE].
R. Emparan, R. Suzuki and K. Tanabe, Quasinormal modes of (Anti-)de Sitter black holes in the 1/D expansion, JHEP 04 (2015) 085 [arXiv:1502.02820] [INSPIRE].
S. Bhattacharyya, A. De, S. Minwalla, R. Mohan and A. Saha, A membrane paradigm at large D, JHEP 04 (2016) 076 [arXiv:1504.06613] [INSPIRE].
H. Liu and M. Mezei, A refinement of entanglement entropy and the number of degrees of freedom, JHEP 04 (2013) 162 [arXiv:1202.2070] [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].
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: 1612.00082
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
Mezei, M. On entanglement spreading from holography. J. High Energ. Phys. 2017, 64 (2017). https://doi.org/10.1007/JHEP05(2017)064
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP05(2017)064