Abstract
We study the evolution of entanglement entropy in a 2-dimensional equilibration process that has a holographic description in terms of a Vaidya geometry. It models a unitary evolution in which the field theory starts in a pure state, its vacuum, and undergoes a perturbation that brings it far from equilibrium. The entanglement entropy in this set up provides a measurement of the quantum entanglement in the system. Using holographic techniques we recover the same result obtained before from the study of processes triggered by a sudden change in a parameter of the hamiltonian, known as quantum quenches. Namely, entanglement in 2-dimensional conformal field theories propagates with velocity v 2 = 1 [1]. Both in quantum quenches and in the Vaidya model equilibration is only achieved at the local level. Remarkably, the holographic derivation of this last fact requires information from behind the apparent horizon generated in the process of gravitational collapse described by the Vaidya geometry. In the early stages of the evolution the apparent horizon seems however to play no relevant role with regard to the entanglement entropy. We speculate on the possibility of deriving a thermalization time for occupation numbers from our analysis.
Similar content being viewed by others
References
P. Calabrese and J .L. Cardy, Evolution of entanglement entropy in one-dimensional systems, J. Stat. Mech. (2005) P04010 [cond-mat/0503393] [SPIRES].
E. Witten, Anti-de Sitter space, thermal phase transition and confinement in gauge theories, Adv. Theor. Math. Phys. 2 (1998) 505 [hep-th/9803131] [SPIRES].
D.T. Son and A.O. Starinets, Viscosity, black holes and quantum field theory, Ann. Rev. Nucl. Part. Sci. 57 (2007) 95 [arXiv:0704.0240] [SPIRES].
P. Kovtun, D.T. Son and A.O. Starinets, Viscosity in strongly interacting quantum field theories from black hole physics, Phys. Rev. Lett. 94 (2005) 111601 [hep-th/0405231] [SPIRES].
R. Baier, P. Romatschke, D.T. Son, A.O. Starinets and M.A. Stephanov, Relativistic viscous hydrodynamics, conformal invariance and holography, JHEP 04 (2008) 100 [arXiv:0712.2451] [SPIRES].
S. Bhattacharyya, V.E. Hubeny, S. Minwalla and M. Rangamani, Nonlinear fluid dynamics from gravity, JHEP 02 (2008) 045 [arXiv:0712.2456] [SPIRES].
P. Reimann, Foundation of statistical mechanics under experimentally realistic conditions, Phys. Rev. Lett. 101 (2008) 190403 [SPIRES].
T. Banks, M.R. Douglas, G.T. Horowitz and E.J. Martinec, AdS dynamics from conformal field theory, hep-th/9808016 [SPIRES].
U.H. Danielsson, E. Keski-Vakkuri and M. Kruczenski, Spherically collapsing matter in AdS, holography and shellons, Nucl. Phys. B 563 (1999) 279 [hep-th/9905227] [SPIRES].
U.H. Danielsson, E. Keski-Vakkuri and M. Kruczenski, Black hole formation in AdS and thermalization on the boundary, JHEP 02 (2000) 039 [hep-th/9912209] [SPIRES].
P.M. Chesler and L.G. Yaffe, Horizon formation and far-from-equilibrium isotropization in supersymmetric Yang-Mills plasma, Phys. Rev. Lett. 102 (2009) 211601 [arXiv:0812.2053] [SPIRES].
G. Beuf, M.P. Heller, R.A. Janik and R. Peschanski, Boost-invariant early time dynamics from AdS/CFT, JHEP 10 (2009) 043 [arXiv:0906.4423] [SPIRES].
P.M. Chesler and L.G. Yaffe, Boost invariant flow, black hole formation and far-fromequilibrium dynamics in N =4 supersymmetric Yang-Mills theory, Phys. Rev. D 82 (2010) 026006 [arXiv:0906.4426] [SPIRES].
S. Bhattacharyya and S. Minwalla, Weak field black hole formation in asymptotically AdS spacetimes, JHEP 09 (2009) 034 [arXiv:0904.0464] [SPIRES].
S.R. Das, T. Nishioka and T. Takayanagi, Probe branes, time-dependent couplings and thermalization in AdS/ CFT , JHEP 07 (2010) 071 [arXiv:1005.3348] [SPIRES].
P. Calabrese and J. Cardy, Entanglement entropy and conformal field theory, J. Phys. A 42 (2009) 504005 [arXiv:0905.4013] [SPIRES].
T. Nishioka, S. Ryu and T. Takayanagi, Holographic entanglement entropy: an overview, J. Phys. A 42 (2009) 504008 [arXiv:0905.0932] [SPIRES].
S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [SPIRES].
S. Ryu and T. Takayanagi, Aspects of holographic entanglement entropy, JHEP 08 (2006) 045 [hep-th/0605073] [SPIRES].
V.E. Hubeny, M. Rangamani and T. Takayanagi, A covariant holographic entanglement entropy proposal, JHEP 07 (2007) 062 [arXiv:0705.0016] [SPIRES].
M.A. Cazalilla, The Luttinger model following a sudden interaction switch-on, Phys. Rev. Lett. 97 (2006) 156403.
A. Iucci and M.A. Cazalilla, Quantum quench dynamics of some exactly solvable models in one dimension, Phys. Rev. A 80 (2009) 063619 [arXiv:0903.1205].
A. Ashtekar and B. Krishnan, Isolated and dynamical horizons and their applications, Living Rev. Rel. 7 (2004) 10 [gr-qc/0407042] [SPIRES].
I. Booth, Black hole boundaries, Can. J. Phys. 83 (2005) 1073 [gr-qc/0508107] [SPIRES].
C. Holzhey, F. Larsen and F. Wilczek, Geometric and renormalized entropy in conformal field theory, Nucl. Phys. B 424 (1994) 443 [hep-th/9403108] [SPIRES].
G. Vidal, J.I. Latorre, E. Rico and A. Kitaev, Entanglement in quantum critical phenomena, Phys. Rev. Lett. 90 (2003) 227902 [quant-ph/0211074] [SPIRES].
J.I. Latorre, E. Rico and G. Vidal, Ground state entanglement in quantum spin chains, Quant. Inf. Comput. 4 (2004) 48 [quant-ph/0304098] [SPIRES].
B.-Q. Jin and V.E. Korepin, Quantum spin chain, Toeplitz determinants and Fisher-Hartwig conjecture, J. of Stat. Phys. 116 (2004) 79 [quant-ph/0304108].
P. Calabrese and J .L. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech. (2004) P06002 [hep-th/0405152] [SPIRES].
S. Sotiriadis and J. Cardy, Inhomogeneous quantum quenches, J. Stat. Mech. (2008) P 11003 [arXiv:0808.0116].
P. Calabrese and J .L. Cardy, Time-dependence of correlation functions following a quantum quench, Phys. Rev. Lett. 96 (2006) 136801 [cond-mat/0601225] [SPIRES].
P. Calabrese and J. Cardy, Quantum quenches in extended systems, J. Stat. Mech. (2007) 06008 [arXiv:0704. 1880] [SPIRES].
G. De Chiara, S. Montangero, P. Calabrese and R. Fazio, Entanglement entropy dynamics in Heisenberg chains, J. Stat. Mech. (2006) P03001 [cond-mat/0512586] [SPIRES].
J. Eisert and T.J. Osborne, General entanglement scaling laws from time evolution, Phys. Rev. Lett. 97 (2006) 150404 [SPIRES].
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].
L. Cincio, J. Dziarmaga, M.M. Rams and W.H. Zurek, Entropy of entanglement and correlations induced by a quench: Dynamics of a quantum phase transition in the quantum Ising model, Phys. Rev. A 75 (2007) [cond-mat/0701768] [SPIRES].
N. Schuch, M.M. Wolf, K.G.H. Vollbrecht and J.I. Cirac, On entropy growth and the hardness of simulating time evolution, New J. Phys. 10 (2008) 033032 [arXiv:0801.2078].
V. Eisler and I. Peschel, Entanglement in a periodic quench, Ann. Phys. (Berlin) 17 (2008) 410 [arXiv:0803. 2655].
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(R) [arXiv:0804.3559].
H. Stephani, D. Kramer, M.A.H. MacCallum, C. Hoenselaers and E. Herlt, Exact solutions of Einstein’s field equations, Cambridge University Press, Cambridge U.K. (2003) pg. 701.
S. Furukawa, V. Pasquier and J. Shiraishi, Mutual information and compactification radius in a c=1 critical phase in one dimension, arXiv:0809.5113 [SPIRES].
P. Calabrese, J. Cardy and E. Tonni, Entanglement entropy of two disjoint intervals in conformal field theory, J. Stat. Mech. (2009) P11001 [arXiv:0905.2069] [SPIRES].
V. Alba, L. Tagliacozzo and P. Calabrese, Entanglement entropy of two disjoint blocks in critical Ising models, arXiv:0910.0706 [SPIRES].
M. Headrick, Entanglement Renyi entropies in holographic theories, arXiv:1006.0047 [SPIRES].
P.C. Vaidya, The external field of a radiating star in general relativity, Curr. Sci. 12 (1943) 183.
P.C. Vaidya, The gravitational field of a radiating star, Pro. Indian Acad. A. 33 (1951) 264.
S.W. Hawking and G.F.R. Ellis, The Large scale structure of space-time, Cambridge University Press, Cambridge U.K. (1973).
V. Balasubramanian and P. Kraus, A stress tensor for anti-de Sitter gravity, Commun. Math. Phys. 208 (1999) 413 [hep-th/9902121] [SPIRES].
S. de Haro, S.N. Solodukhin and K. Skenderis, Holographic reconstruction of spacetime and renormalization in the AdS/CFT correspondence, Commun. Math. Phys. 217 (2001) 595 [hep-th/0002230] [SPIRES].
M. Srednicki, Entropy and area, Phys. Rev. Lett. 71 (1993) 666 [hep-th/9303048] [SPIRES].
P. Figueras, V.E. Hubeny, M. Rangamani and S.F. Ross, Dynamical black holes and expanding plasmas, JHEP 04 (2009) 137 [arXiv:0902.4696] [SPIRES].
D.V. Fursaev, Proof of the holographic formula for entanglement entropy, JHEP 09 (2006) 018 [hep-th/0606184] [SPIRES].
M. Cramer, C. M. Dawson, J. Eisert and T.J. Osborne, Exact relaxation in a class of nonequilibrium quantum lattice systems, Phys. Rev. Lett. 100 (2008) 030602 [cond-mat/0703314] [SPIRES].
V. Balasubramanian and S.F. Ross, Holographic particle detection, Phys. Rev. D 61 (2000) 044007 [hep-th/9906226] [SPIRES].
I. Amado, C. Hoyos-Badajoz, K. Landsteiner and S. Montero, Hydrodynamics and beyond in the strongly coupled N =4 plasma, JHEP 07 (2008) 133 [arXiv:0805.2570] [SPIRES].
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint:1006.4090
Rights and permissions
About this article
Cite this article
Abajo-Arrastia, J., Aparício, J. & López, E. Holographic evolution of entanglement entropy. J. High Energ. Phys. 2010, 149 (2010). https://doi.org/10.1007/JHEP11(2010)149
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP11(2010)149