Nonlocal probes of thermalization in holographic quenches with spectral methods. (English) Zbl 1388.83092
Summary: We describe the application of pseudo-spectral methods to problems of holographic thermal quenches of relevant couplings in strongly coupled gauge theories. We focus on quenches of a fermionic mass term in a strongly coupled \( \mathcal{N}=4 \) supersymmetric Yang-Mills plasma, and the subsequent equilibration of the system. From the dual gravitational perspective, we study gravitational collapse of a massive scalar field in asymptotically anti-de-Sitter geometry with a prescribed boundary condition for its non-normalizable mode. Access to the full background geometry of the gravitational collapse allows for the study of nonlocal probes of the thermalization process. We discuss the evolution of the apparent and the event horizons, the two-point correlation functions of operators of large conformal dimensions, and the evolution of the entanglement entropy of the system. We compare the thermalization process from the viewpoint of local (the one-point) correlation functions and these nonlocal probes, finding that the thermalization time as measured by the probes is length dependent, and can exceed that of the one-point function. We further discuss how the different energy scales of the problem contribute to its thermalization.
MSC:
| 83C45 | Quantization of the gravitational field |
Keywords:
gauge-gravity correspondence; AdS-CFT correspondence; holography and condensed matter physics (AdS/CMT)References:
| [1] | M. Greiner, O. Mandel, T. W. Hänsch and I. Bloch, Collapse and revival of the matter wave field of a Bose-Einstein condensate, Nature419 (2002) 51. |
| [2] | T. Kinoshita, T. Wenger and D.S. Weiss, A quantum Newton’s cradle, Nature440 (2006) 900. |
| [3] | S. Hofferberth, I. Lesanovsky, B. Fischer, T. Schumm and J. Schmiedmayer, Non-equilibrium coherence dynamics in one-dimensional Bose gases, Nature449 (2007) 324. |
| [4] | S. Will et al., Time-resolved observation of coherent multi-body interactions in quantum phase revivals, Nature465 (2010) 197. |
| [5] | S. Trotzky et al., Probing the relaxation towards equilibrium in an isolated strongly correlated one-dimensional Bose gas, Nature Phys.8 (2012) 325. |
| [6] | S. Mondal, D. Sen and K. Sengupta, Non-equilibrium dynamics of quantum systems: order parameter evolution, defect generation, and qubit transfer in Quantum quenching, anealing and computation, A. Das et al. eds., Lect. Notes Phys.802 (2010) 21 [arXiv:0908.2922]. · Zbl 1201.81031 |
| [7] | J. Dziarmaga, Dynamics of a quantum phase transition and relaxation to a steady state, Adv. Phys.59 (2010) 1063 [arXiv:0912.4034]. |
| [8] | A. Polkovnikov, K. Sengupta, A. Silva and M. Vengalattore, Nonequilibrium dynamics of closed interacting quantum systems, Rev. Mod. Phys.83 (2011) 863 [arXiv:1007.5331] [INSPIRE]. |
| [9] | A. Lamacraft and J.E. Moore, Potential insights into non-equilibrium behavior from atomic physics, in Ultracold bosonic and fermionic gases, A. Flethcer et al. eds., Contemporary Concepts in Condensed Matter Science, Elsevier (2012), arXiv:1106.3567. |
| [10] | P. Calabrese and J.L. Cardy, Time-dependence of correlation functions following a quantum quench, Phys. Rev. Lett.96 (2006) 136801 [cond-mat/0601225] [INSPIRE]. |
| [11] | P. Calabrese and J. Cardy, Quantum quenches in extended systems, J. Stat. Mech.0706 (2007) P06008 [arXiv:0704.1880] [INSPIRE]. · Zbl 1456.81358 |
| [12] | P. Calabrese, C. Hagendorf and P. Le Doussal, Time evolution of 1D gapless models from a domain-wall initial state: SLE continued?, J. Stat. Mech. (2008) P07013 [arXiv:0804.2431]. |
| [13] | S. Sotiriadis and J. Cardy, Inhomogeneous quantum quenches, J. Stat. Mech. (2008) P11003 [arXiv:0808.0116]. |
| [14] | P. Calabrese and J.L. Cardy, Evolution of entanglement entropy in one-dimensional systems, J. Stat. Mech. (2005) P04010 [cond-mat/0503393] [INSPIRE]. · Zbl 1456.82578 |
| [15] | C. Kollath, A. Laeuchli and E. Altman, Quench dynamics and nonequilibrium phase diagram of the bose-hubbard model, Phys. Rev. Lett.98 (2006) 180601. |
| [16] | M. Cramer, C. M. Dawson, J. Eisert and T.J. Osborne, Exact relaxation in a class of non-equilibrium quantum lattice systems, Phys. Rev. Lett.100 (2008) 030602 [cond-mat/0703314]. |
| [17] | S. Sotiriadis and J. Cardy, Inhomogeneous quantum quenches, J. Stat. Mech. (2008) P11003 [arXiv:0808.0116]. |
| [18] | G. Roux, Quenches in quantum many-body systems: one-dimensional Bose-Hubbard model reexamined, Phys. Rev.A 79 (2009) 021608 [arXiv:0810.3720]. |
| [19] | S. Sotiriadis, P. Calabrese and J. Cardy, Quantum quench from a thermal initial state, EPL87 (2009) 20002 [arXiv:0903.0895]. |
| [20] | M. Rigol, V. Dunjko, V. Yurovsky and M. Olshanii, Relaxation in a completely integrable many-body quantum system: an ab initio study of the dynamics of the highly excited states of lattice hard-core bosons, Phys. Rev. Lett.98 (2007) 050405 [cond-mat/0604476]. |
| [21] | S. R. Manmana, S. Wessel, R.M. Noack and A. Muramatsu, Strongly correlated fermions after a quantum quench, Phys. Rev. Lett.98 (2006) 4 [cond-mat/0612030]. |
| [22] | M. Rigol, V. Dunjko and M. Olshanii, Thermalization and its mechanism for generic isolated quantum systems, Nature452 (2008) 854 [arXiv:0708.1324]. |
| [23] | P. Calabrese, F.H.L. Essler and M. Fagotti, Quantum quench in the transverse field ising chain, Phys. Rev. Lett.106 (2011) 227203 [arXiv:1104.0154]. |
| [24] | S. Sotiriadis and J. Cardy, Quantum quench in interacting field theory: A Self-consistent approximation, Phys. Rev.B 81 (2010) 134305 [arXiv:1002.0167] [INSPIRE]. |
| [25] | S.R. Das and K. Sengupta, Non-equilibrium dynamics of O(N ) nonlinear σ-models: a large-N approach, JHEP09 (2012) 072 [arXiv:1202.2458] [INSPIRE]. |
| [26] | L.-Y. Hung, M. Smolkin and E. Sorkin, Modification of late time phase structure by quantum quenches, Phys. Rev. Lett.109 (2012) 155702 [arXiv:1206.2685] [INSPIRE]. |
| [27] | L.-Y. Hung, M. Smolkin and E. Sorkin, (Non) supersymmetric quantum quenches, JHEP12 (2013) 022 [arXiv:1307.0376] [INSPIRE]. · Zbl 1396.81104 |
| [28] | 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]. · Zbl 0914.53047 |
| [29] | O. Aharony, S.S. Gubser, J.M. Maldacena, H. Ooguri and Y. Oz, Large-N field theories, string theory and gravity, Phys. Rept.323 (2000) 183 [hep-th/9905111] [INSPIRE]. · Zbl 1368.81009 |
| [30] | 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] [INSPIRE]. · Zbl 0953.83038 |
| [31] | U.H. Danielsson, E. Keski-Vakkuri and M. Kruczenski, Black hole formation in AdS and thermalization on the boundary, JHEP02 (2000) 039 [hep-th/9912209] [INSPIRE]. · Zbl 0959.83024 |
| [32] | S.B. Giddings and S.F. Ross, D3-brane shells to black branes on the Coulomb branch, Phys. Rev.D 61 (2000) 024036 [hep-th/9907204] [INSPIRE]. |
| [33] | S.B. Giddings and A. Nudelman, Gravitational collapse and its boundary description in AdS, JHEP02 (2002) 003 [hep-th/0112099] [INSPIRE]. |
| [34] | R.A. Janik and R.B. Peschanski, Gauge/gravity duality and thermalization of a boost-invariant perfect fluid, Phys. Rev.D 74 (2006) 046007 [hep-th/0606149] [INSPIRE]. |
| [35] | R.A. Janik, Viscous plasma evolution from gravity using AdS/CFT, Phys. Rev. Lett.98 (2007) 022302 [hep-th/0610144] [INSPIRE]. |
| [36] | S. Lin and E. Shuryak, Toward the AdS/CFT gravity dual for high energy collisions. 3. Gravitationally collapsing shell and quasiequilibrium, Phys. Rev.D 78 (2008) 125018 [arXiv:0808.0910] [INSPIRE]. |
| [37] | S. Bhattacharyya and S. Minwalla, Weak field black hole formation in asymptotically AdS spacetimes, JHEP09 (2009) 034 [arXiv:0904.0464] [INSPIRE]. |
| [38] | 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] [INSPIRE]. |
| [39] | P.M. Chesler and L.G. Yaffe, Boost invariant flow, black hole formation and far-from-equilibrium dynamics in N = 4 supersymmetric Yang-Mills theory, Phys. Rev.D 82 (2010) 026006 [arXiv:0906.4426] [INSPIRE]. |
| [40] | S.R. Das, T. Nishioka and T. Takayanagi, Probe branes, time-dependent couplings and thermalization in AdS/CFT, JHEP07 (2010) 071 [arXiv:1005.3348] [INSPIRE]. · Zbl 1290.81090 |
| [41] | D. Garfinkle and L.A. Pando Zayas, Rapid thermalization in field theory from gravitational collapse, Phys. Rev.D 84 (2011) 066006 [arXiv:1106.2339] [INSPIRE]. |
| [42] | D. Garfinkle, L.A. Pando Zayas and D. Reichmann, On field theory thermalization from gravitational collapse, JHEP02 (2012) 119 [arXiv:1110.5823] [INSPIRE]. · Zbl 1309.81272 |
| [43] | S.R. Das, Holographic quantum quench, J. Phys. Conf. Ser.343 (2012) 012027 [arXiv:1111.7275] [INSPIRE]. |
| [44] | H. Bantilan, F. Pretorius and S.S. Gubser, Simulation of asymptotically AdS5Spacetimes with a Generalized Harmonic Evolution Scheme, Phys. Rev.D 85 (2012) 084038 [arXiv:1201.2132] [INSPIRE]. |
| [45] | M.P. Heller, D. Mateos, W. van der Schee and D. Trancanelli, Strong coupling isotropization of non-abelian plasmas simplified, Phys. Rev. Lett.108 (2012) 191601 [arXiv:1202.0981] [INSPIRE]. |
| [46] | M.P. Heller, R.A. Janik and P. Witaszczyk, A numerical relativity approach to the initial value problem in asymptotically Anti-de Sitter spacetime for plasma thermalization — An ADM formulation, Phys. Rev.D 85 (2012) 126002 [arXiv:1203.0755] [INSPIRE]. |
| [47] | M.J. Bhaseen, J.P. Gauntlett, B.D. Simons, J. Sonner and T. Wiseman, Holographic superfluids and the dynamics of symmetry breaking, Phys. Rev. Lett.110 (2013) 015301 [arXiv:1207.4194] [INSPIRE]. |
| [48] | B. Wu, On holographic thermalization and gravitational collapse of massless scalar fields, JHEP10 (2012) 133 [arXiv:1208.1393] [INSPIRE]. |
| [49] | B. Wu, On holographic thermalization and gravitational collapse of tachyonic scalar fields, JHEP04 (2013) 044 [arXiv:1301.3796] [INSPIRE]. |
| [50] | E. Caceres, A. Kundu, J.F. Pedraza and D.L. Yang, Weak field collapse in AdS: introducing a charge density, arXiv:1411.1744. · Zbl 1388.83195 |
| [51] | A. Buchel, L. Lehner and R.C. Myers, Thermal quenches in N = 2* plasmas, JHEP08 (2012) 049 [arXiv:1206.6785] [INSPIRE]. |
| [52] | A. Buchel, L. Lehner, R.C. Myers and A. van Niekerk, Quantum quenches of holographic plasmas, JHEP05 (2013) 067 [arXiv:1302.2924] [INSPIRE]. |
| [53] | A. Buchel, R.C. Myers and A. van Niekerk, Universality of abrupt holographic quenches, Phys. Rev. Lett.111 (2013) 201602 [arXiv:1307.4740] [INSPIRE]. |
| [54] | M. Nozaki, T. Numasawa and T. Takayanagi, Holographic local quenches and entanglement density, JHEP05 (2013) 080 [arXiv:1302.5703] [INSPIRE]. · Zbl 1342.83111 |
| [55] | T. Hartman and J. Maldacena, Time evolution of entanglement entropy from black hole interiors, JHEP05 (2013) 014 [arXiv:1303.1080] [INSPIRE]. · Zbl 1342.83170 |
| [56] | N. Engelhardt and G.T. Horowitz, Entanglement entropy near cosmological singularities, JHEP06 (2013) 041 [arXiv:1303.4442] [INSPIRE]. · Zbl 1342.83238 |
| [57] | P. Basu and A. Ghosh, Dissipative nonlinear dynamics in holography, Phys. Rev.D 89 (2014) 046004 [arXiv:1304.6349] [INSPIRE]. |
| [58] | W.-J. Li, Y. Tian and H.-b. Zhang, Periodically driven holographic superconductor, JHEP07 (2013) 030 [arXiv:1305.1600] [INSPIRE]. |
| [59] | Y.-Z. Li, S.-F. Wu, Y.-Q. Wang and G.-H. Yang, Linear growth of entanglement entropy in holographic thermalization captured by horizon interiors and mutual information, JHEP09 (2013) 057 [arXiv:1306.0210] [INSPIRE]. |
| [60] | K. Hashimoto and T. Oka, Vacuum instability in electric fields via AdS/CFT: Euler-Heisenberg lagrangian and planckian thermalization, JHEP10 (2013) 116 [arXiv:1307.7423] [INSPIRE]. |
| [61] | R. Auzzi, S. Elitzur, S.B. Gudnason and E. Rabinovici, On periodically driven AdS/CFT, JHEP11 (2013) 016 [arXiv:1308.2132] [INSPIRE]. |
| [62] | P. Basu, D. Das, S.R. Das and K. Sengupta, Quantum quench and double trace couplings, JHEP12 (2013) 070 [arXiv:1308.4061] [INSPIRE]. |
| [63] | P.M. Chesler and L.G. Yaffe, Numerical solution of gravitational dynamics in asymptotically anti-de Sitter spacetimes, JHEP07 (2014) 086 [arXiv:1309.1439] [INSPIRE]. · Zbl 1421.81111 |
| [64] | X.-X. Zeng, X.-M. Liu and W.-B. Liu, Holographic thermalization with a chemical potential in Gauss-Bonnet gravity, JHEP03 (2014) 031 [arXiv:1311.0718] [INSPIRE]. |
| [65] | T. Ugajin, Two dimensional quantum quenches and holography, arXiv:1311.2562 [INSPIRE]. · Zbl 1437.83118 |
| [66] | X.O. Camanho, J.D. Edelstein, G. Giribet and A. Gomberoff, Generalized phase transitions in Lovelock gravity, Phys. Rev.D 90 (2014) 064028 [arXiv:1311.6768] [INSPIRE]. |
| [67] | J.F. Pedraza, Evolution of nonlocal observables in an expanding boost-invariant plasma, Phys. Rev.D 90 (2014) 046010 [arXiv:1405.1724] [INSPIRE]. |
| [68] | X. Bai, B.-H. Lee, M. Park and K. Sunly, Dynamical condensation in a holographic superconductor model with anisotropy, JHEP09 (2014) 054 [arXiv:1405.1806] [INSPIRE]. |
| [69] | A.F. Astaneh and A.E. Mosaffa, Quantum local quench, AdS/BCFT and yo-yo string, arXiv:1405.5469 [INSPIRE]. · Zbl 1388.83166 |
| [70] | M. Nozaki, Notes on quantum entanglement of local operators, JHEP10 (2014) 147 [arXiv:1405.5875] [INSPIRE]. · Zbl 1333.81293 |
| [71] | D. Berenstein and A. Miller, Conformal perturbation theory, dimensional regularization and AdS/CFT correspondence, Phys. Rev.D 90 (2014) 086011 [arXiv:1406.4142] [INSPIRE]. |
| [72] | K. Hashimoto, S. Kinoshita, K. Murata and T. Oka, Electric field quench in AdS/CFT, JHEP09 (2014) 126 [arXiv:1407.0798] [INSPIRE]. |
| [73] | V. Cardoso, L. Gualtieri, C. Herdeiro and U. Sperhake, Exploring new physics frontiers through numerical relativity, arXiv:1409.0014 [INSPIRE]. · Zbl 1347.83017 |
| [74] | V. Balasubramanian et al., Thermalization of strongly coupled field theories, Phys. Rev. Lett.106 (2011) 191601 [arXiv:1012.4753] [INSPIRE]. |
| [75] | V. Balasubramanian et al., Holographic thermalization, Phys. Rev.D 84 (2011) 026010 [arXiv:1103.2683] [INSPIRE]. |
| [76] | J. Abajo-Arrastia, J. Aparicio and E. Lopez, Holographic evolution of entanglement entropy, JHEP11 (2010) 149 [arXiv:1006.4090] [INSPIRE]. · Zbl 1294.81128 |
| [77] | H. Ebrahim and M. Headrick, Instantaneous thermalization in holographic plasmas, arXiv:1010.5443 [INSPIRE]. |
| [78] | V. Balasubramanian, A. Bernamonti, N. Copland, B. Craps and F. Galli, Thermalization of mutual and tripartite information in strongly coupled two dimensional conformal field theories, Phys. Rev.D 84 (2011) 105017 [arXiv:1110.0488] [INSPIRE]. |
| [79] | 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]. · Zbl 1448.83015 |
| [80] | J. Aparicio and E. Lopez, Evolution of two-point functions from holography, JHEP12 (2011) 082 [arXiv:1109.3571] [INSPIRE]. · Zbl 1306.81145 |
| [81] | A. Allais and E. Tonni, Holographic evolution of the mutual information, JHEP01 (2012) 102 [arXiv:1110.1607] [INSPIRE]. · Zbl 1306.81144 |
| [82] | V. Keranen, E. Keski-Vakkuri and L. Thorlacius, Thermalization and entanglement following a non-relativistic holographic quench, Phys. Rev.D 85 (2012) 026005 [arXiv:1110.5035] [INSPIRE]. |
| [83] | D. Galante and M. Schvellinger, Thermalization with a chemical potential from AdS spaces, JHEP07 (2012) 096 [arXiv:1205.1548] [INSPIRE]. |
| [84] | E. Caceres and A. Kundu, Holographic thermalization with chemical potential, JHEP09 (2012) 055 [arXiv:1205.2354] [INSPIRE]. |
| [85] | I.Y. Arefeva and I.V. Volovich, On holographic thermalization and dethermalization of quark-gluon plasma, arXiv:1211.6041 [INSPIRE]. |
| [86] | W. Baron, D. Galante and M. Schvellinger, Dynamics of holographic thermalization, JHEP03 (2013) 070 [arXiv:1212.5234] [INSPIRE]. |
| [87] | V. Balasubramanian et al., Inhomogeneous holographic thermalization, JHEP10 (2013) 082 [arXiv:1307.7086] [INSPIRE]. |
| [88] | P. Fonda et al., Holographic thermalization with Lifshitz scaling and hyperscaling violation, JHEP08 (2014) 051 [arXiv:1401.6088] [INSPIRE]. |
| [89] | X.-X. Zeng, X.-M. Liu and W.-B. Liu, Holographic thermalization in noncommutative geometry, arXiv:1407.5262 [INSPIRE]. · Zbl 1330.81134 |
| [90] | X.-X. Zeng, D.-Y. Chen and L.-F. Li, Holographic thermalization and gravitational collapse in the spacetime dominated by quintessence dark energy, arXiv:1408.6632 [INSPIRE]. |
| [91] | H. Liu and S.J. Suh, Entanglement tsunami: universal scaling in holographic thermalization, Phys. Rev. Lett.112 (2014) 011601 [arXiv:1305.7244] [INSPIRE]. |
| [92] | H. Liu and S.J. Suh, Entanglement growth during thermalization in holographic systems, Phys. Rev.D 89 (2014) 066012 [arXiv:1311.1200] [INSPIRE]. |
| [93] | P. Basu and S.R. Das, Quantum quench across a holographic critical point, JHEP01 (2012) 103 [arXiv:1109.3909] [INSPIRE]. · Zbl 1306.81191 |
| [94] | P. Basu, D. Das, S.R. Das and T. Nishioka, Quantum quench across a zero temperature holographic superfluid transition, JHEP03 (2013) 146 [arXiv:1211.7076] [INSPIRE]. · Zbl 1342.83229 |
| [95] | P.M. Chesler and L.G. Yaffe, Holography and colliding gravitational shock waves in asymptotically AdS5spacetime, Phys. Rev. Lett.106 (2011) 021601 [arXiv:1011.3562] [INSPIRE]. |
| [96] | W. van der Schee, P. Romatschke and S. Pratt, Fully dynamical simulation of central nuclear collisions, Phys. Rev. Lett.111 (2013) 222302 [arXiv:1307.2539] [INSPIRE]. |
| [97] | J. Casalderrey-Solana, M.P. Heller, D. Mateos and W. van der Schee, Longitudinal coherence in a holographic model of asymmetric collisions, Phys. Rev. Lett.112 (2014) 221602 [arXiv:1312.2956] [INSPIRE]. |
| [98] | D. Fernández, Towards collisions of inhomogeneous shockwaves in AdS, arXiv:1407.5628 [INSPIRE]. · Zbl 1388.83077 |
| [99] | A. Buchel and J.T. Liu, Thermodynamics of the N = 2∗flow, JHEP11 (2003) 031 [hep-th/0305064] [INSPIRE]. |
| [100] | A. Cherman and A. Nellore, Universal relations of transport coefficients from holography, Phys. Rev.D 80 (2009) 066006 [arXiv:0905.2969] [INSPIRE]. |
| [101] | J.P. Boyd, Chebyshev and Fourier spectral methods, second edition, Dover Publications Inc. (2000). |
| [102] | K. Pilch and N.P. Warner, N = 2 supersymmetric RG flows and the IIB dilaton, Nucl. Phys.B 594 (2001) 209 [hep-th/0004063] [INSPIRE]. · Zbl 0971.83513 |
| [103] | A. Khavaev, K. Pilch and N.P. Warner, New vacua of gauged N = 8 supergravity in five-dimensions, Phys. Lett.B 487 (2000) 14 [hep-th/9812035] [INSPIRE]. · Zbl 1050.81686 |
| [104] | A. Buchel, A.W. Peet and J. Polchinski, Gauge dual and noncommutative extension of an N =2 supergravity solution, Phys. Rev.D 63(2001) 044009 [hep-th/0008076] [INSPIRE]. |
| [105] | N.J. Evans, C.V. Johnson and M. Petrini, The enhancon and N = 2 gauge theory: gravity RG flows, JHEP10 (2000) 022 [hep-th/0008081] [INSPIRE]. · Zbl 0965.81050 |
| [106] | E. D’Hoker and D.Z. Freedman, Supersymmetric gauge theories and the AdS/CFT correspondence, hep-th/0201253 [INSPIRE]. |
| [107] | P.C. Vaidya, The external field of a radiating star in general relativity, Curr. Sci.12 (1943) 183. |
| [108] | E. Poisson, A relativist’s toolkit, Cambridge University Press, Cambridge U.K. (2004). · Zbl 1058.83002 |
| [109] | J.D. Bekenstein, Black holes and entropy, Phys. Rev.D 7 (1973) 2333 [INSPIRE]. · Zbl 1369.83037 |
| [110] | S.W. Hawking, Particle creation by black holes, Commun. Math. Phys.43 (1975) 199 [Erratum ibid.46 (1976) 206] [INSPIRE]. · Zbl 1378.83040 |
| [111] | V.E. Hubeny, M. Rangamani and T. Takayanagi, A covariant holographic entanglement entropy proposal, JHEP07 (2007) 062 [arXiv:0705.0016] [INSPIRE]. |
| [112] | S. Bhattacharyya et al., Local fluid dynamical entropy from gravity, JHEP06 (2008) 055 [arXiv:0803.2526] [INSPIRE]. |
| [113] | V. Balasubramanian and S.F. Ross, Holographic particle detection, Phys. Rev.D 61 (2000) 044007 [hep-th/9906226] [INSPIRE]. |
| [114] | J. Louko, D. Marolf and S.F. Ross, On geodesic propagators and black hole holography, Phys. Rev.D 62 (2000) 044041 [hep-th/0002111] [INSPIRE]. |
| [115] | S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett.96 (2006) 181602 [hep-th/0603001] [INSPIRE]. · Zbl 1228.83110 |
| [116] | S. Ryu and T. Takayanagi, Aspects of holographic entanglement entropy, JHEP08 (2006) 045 [hep-th/0605073] [INSPIRE]. |
| [117] | L.-Y. Hung, R.C. Myers and M. Smolkin, Some calculable contributions to holographic entanglement entropy, JHEP08 (2011) 039 [arXiv:1105.6055] [INSPIRE]. · Zbl 1298.81216 |
| [118] | A. Lewkowycz, R.C. Myers and M. Smolkin, Observations on entanglement entropy in massive QFT’s, JHEP04 (2013) 017 [arXiv:1210.6858] [INSPIRE]. · Zbl 1342.81364 |
| [119] | H. Liu and M. Mezei, A refinement of entanglement entropy and the number of degrees of freedom, JHEP04 (2013) 162 [arXiv:1202.2070] [INSPIRE]. · Zbl 1342.81346 |
| [120] | R.C. Myers and A. Singh, Comments on holographic entanglement entropy and RG flows, JHEP04 (2012) 122 [arXiv:1202.2068] [INSPIRE]. · Zbl 1348.81337 |
| [121] | T. Nishioka, S. Ryu and T. Takayanagi, Holographic entanglement entropy: an overview, J. Phys.A 42 (2009) 504008 [arXiv:0905.0932] [INSPIRE]. · Zbl 1179.81138 |
| [122] | L. Lehner, Numerical relativity: a review, Class. Quant. Grav.18 (2001) R25 [gr-qc/0106072] [INSPIRE]. · Zbl 0987.83001 |
| [123] | H.P. Pfeiffer, L.E. Kidder, M.A. Scheel and S.A. Teukolsky, A multidomain spectral method for solving elliptic equations, Comput. Phys. Commun.152 (2003) 253 [gr-qc/0202096] [INSPIRE]. · Zbl 1196.65179 |
| [124] | S.R. Das, D.A. Galante and R.C. Myers, Universal scaling in fast quantum quenches in conformal field theories, Phys. Rev. Lett.112 (2014) 171601 [arXiv:1401.0560] [INSPIRE]. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
