Abstract
We study the non-equilibrium condensation process in a holographic superconductor with nonlinear corrections to the U (1) gauge field. We start with an asymptotic Anti-de-Sitter (AdS) black hole against a complex scalar perturbation at the initial time, and solve the dynamics of the gravitational systems in the bulk. When the black hole temperature T is smaller than a critical value T c , the scalar perturbation grows exponentially till saturation, the final state of spacetime approaches to a hairy black hole. In the bulk theory, we find the clue of the influence of nonlinear corrections in the gauge filed on the process of the scalar field condensation. We show that the bulk dynamics in the non-equilibrium process is completely consistent with the observations on the boundary order parameter. Furthermore we examine the time evolution of horizons in the bulk non-equilibrium transformation process from the bald AdS black hole to the AdS hairy hole. Both the evolution of apparent and event horizons show that the original AdS black hole configuration requires more time to finish the transformation to become a hairy black hole if there is nonlinear correction to the electromagnetic field. We generalize our non-equilibrium discussions to the holographic entanglement entropy and find that the holographic entanglement entropy can give us further understanding of the influence of the nonlinearity in the gauge field on the scalar condensation.
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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.A. Hartnoll, Lectures on holographic methods for condensed matter physics, Class. Quant. Grav. 26 (2009) 224002 [arXiv:0903.3246] [INSPIRE].
G.T. Horowitz, Introduction to Holographic Superconductors, Lect. Notes Phys. 828 (2011) 313 [arXiv:1002.1722] [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].
S.A. Hartnoll, C.P. Herzog and G.T. Horowitz, Building a Holographic Superconductor, Phys. Rev. Lett. 101 (2008) 031601 [arXiv:0803.3295] [INSPIRE].
S.S. Gubser, Breaking an Abelian gauge symmetry near a black hole horizon, Phys. Rev. D 78 (2008) 065034 [arXiv:0801.2977] [INSPIRE].
S.A. Hartnoll, C.P. Herzog and G.T. Horowitz, Holographic Superconductors, JHEP 12 (2008) 015 [arXiv:0810.1563] [INSPIRE].
C.P. Herzog, Lectures on Holographic Superfluidity and Superconductivity, J. Phys. A 42 (2009) 343001 [arXiv:0904.1975] [INSPIRE].
G.T. Horowitz and M.M. Roberts, Holographic Superconductors with Various Condensates, Phys. Rev. D 78 (2008) 126008 [arXiv:0810.1077] [INSPIRE].
E. Nakano and W.-Y. Wen, Critical magnetic field in a holographic superconductor, Phys. Rev. D 78 (2008) 046004 [arXiv:0804.3180] [INSPIRE].
G. Koutsoumbas, E. Papantonopoulos and G. Siopsis, Exact Gravity Dual of a Gapless Superconductor, JHEP 07 (2009) 026 [arXiv:0902.0733] [INSPIRE].
J. Sonner, A Rotating Holographic Superconductor, Phys. Rev. D 80 (2009) 084031 [arXiv:0903.0627] [INSPIRE].
S.S. Gubser, C.P. Herzog, S.S. Pufu and T. Tesileanu, Superconductors from Superstrings, Phys. Rev. Lett. 103 (2009) 141601 [arXiv:0907.3510] [INSPIRE].
Y. Liu, Q. Pan and B. Wang, Holographic superconductor developed in BTZ black hole background with backreactions, Phys. Lett. B 702 (2011) 94 [arXiv:1106.4353] [INSPIRE].
Y. Liu, Q. Pan and B. Wang, Holographic superconductor developed in BTZ black hole background with backreactions, Phys. Lett. B 702 (2011) 94 [arXiv:1106.4353] [INSPIRE].
J.P. Gauntlett, J. Sonner and T. Wiseman, Holographic superconductivity in M-theory, Phys. Rev. Lett. 103 (2009) 151601 [arXiv:0907.3796] [INSPIRE].
J. Jing and S. Chen, Holographic superconductors in the Born-Infeld electrodynamics, Phys. Lett. B 686 (2010) 68 [arXiv:1001.4227] [INSPIRE].
C.P. Herzog, An Analytic Holographic Superconductor, Phys. Rev. D 81 (2010) 126009 [arXiv:1003.3278] [INSPIRE].
K. Maeda, M. Natsuume and T. Okamura, Universality class of holographic superconductors, Phys. Rev. D 79 (2009) 126004 [arXiv:0904.1914] [INSPIRE].
R. Gregory, S. Kanno and J. Soda, Holographic Superconductors with Higher Curvature Corrections, JHEP 10 (09) 010 [arXiv:0907.3203] [INSPIRE].
X.-H. Ge, B. Wang, S.-F. Wu and G.-H. Yang, Analytical study on holographic superconductors in external magnetic field, JHEP 08 (2010) 108 [arXiv:1002.4901] [INSPIRE].
Y. Brihaye and B. Hartmann, Holographic Superconductors in 3+1 dimensions away from the probe limit, Phys. Rev. D 81 (2010) 126008 [arXiv:1003.5130] [INSPIRE].
Y. Brihaye and B. Hartmann, Holographic superfluid/fluid/insulator phase transitions in 2+1 dimensions, Phys. Rev. D 83 (2011) 126008 [arXiv:1101.5708] [INSPIRE].
G.T. Horowitz and B. Way, Complete Phase Diagrams for a Holographic Superconductor/Insulator System, JHEP 11 (2010) 011 [arXiv:1007.3714] [INSPIRE].
S. Franco, A. Garcia-Garcia and D. Rodriguez-Gomez, A General class of holographic superconductors, JHEP 04 (2010) 092 [arXiv:0906.1214] [INSPIRE].
S. Franco, A.M. Garcia-Garcia and D. Rodriguez-Gomez, A holographic approach to phase transitions, Phys. Rev. D 81 (2010) 041901 [arXiv:0911.1354] [INSPIRE].
Q. Pan and B. Wang, General holographic superconductor models with Gauss-Bonnet corrections, Phys. Lett. B 693 (2010) 159 [arXiv:1005.4743] [INSPIRE].
Y. Peng and Q.Y. Pan, Stückelberg Holographic Superconductor Models with Backreactions, Commun. Theor. Phys. 59 (2013) 110.
Y. Peng, Q. Pan and B. Wang, Various types of phase transitions in the AdS soliton background, Phys. Lett. B 699 (2011) 383 [arXiv:1104.2478] [INSPIRE].
C.P. Herzog, P.K. Kovtun and D.T. Son, Holographic model of superfluidity, Phys. Rev. D 79 (2009) 066002 [arXiv:0809.4870] [INSPIRE].
Y. Peng and Y. Liu, A general holographic metal/superconductor phase transition model, JHEP 02 (2015) 082 [arXiv:1410.7234] [INSPIRE].
K. Murata, S. Kinoshita and N. Tanahashi, Non-equilibrium Condensation Process in a Holographic Superconductor, JHEP 07 (2010) 050 [arXiv:1005.0633] [INSPIRE].
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].
X. Bai, B.-H. Lee, M. Park and K. Sunly, Dynamical Condensation in a Holographic Superconductor Model with Anisotropy, JHEP 09 (2014) 054 [arXiv:1405.1806] [INSPIRE].
W.-J. Li, Y. Tian and H.-b. Zhang, Periodically Driven Holographic Superconductor, JHEP 07 (2013) 030 [arXiv:1305.1600] [INSPIRE].
X. Gao, A.M. Garcia-Garcia, H.B. Zeng and H.-Q. Zhang, Normal modes and time evolution of a holographic superconductor after a quantum quench, JHEP 06 (2014) 019 [arXiv:1212.1049] [INSPIRE].
A.M. Garcıa-García, H.B. Zeng and H.Q. Zhang, A thermal quench induces spatial inhomogeneities in a holographic superconductor, JHEP 07 (2014) 096 [arXiv:1308.5398] [INSPIRE].
A. Buchel, R.C. Myers and A. van Niekerk, Nonlocal probes of thermalization in holographic quenches with spectral methods, JHEP 02 (2015) 017 [Erratum ibid. 1507 (2015) 137] [arXiv:1410.6201] [INSPIRE].
X.-X. Zeng, X.-Y. Hu and L.-F. Li, Effect of phantom dark energy on the holographic thermalization, arXiv:1503.04518 [INSPIRE].
T. Ishii, E. Kiritsis and C. Rosen, Thermalization in a Holographic Confining Gauge Theory, JHEP 08 (2015) 008 [arXiv:1503.07766] [INSPIRE].
J. Sonner, A. del Campo and W.H. Zurek, Universal far-from-equilibrium Dynamics of a Holographic Superconductor, arXiv:1406.2329 [INSPIRE].
P.M. Chesler, A.M. Garcia-Garcia and H. Liu, Defect Formation beyond Kibble-Zurek Mechanism and Holography, Phys. Rev. X 5 (2015) 021015 [arXiv:1407.1862] [INSPIRE].
L. Barclay, R. Gregory, S. Kanno and P. Sutcliffe, Gauss-Bonnet Holographic Superconductors, JHEP 12 (2010) 029 [arXiv:1009.1991] [INSPIRE].
M. Siani, Holographic Superconductors and Higher Curvature Corrections, JHEP 12 (2010) 035 [arXiv:1010.0700] [INSPIRE].
J. Jing, L. Wang, Q. Pan and S. Chen, Holographic Superconductors in Gauss-Bonnet gravity with Born-Infeld electrodynamics, Phys. Rev. D 83 (2011) 066010 [arXiv:1012.0644] [INSPIRE].
R. Gregory, Holographic Superconductivity with Gauss-Bonnet gravity, J. Phys. Conf. Ser. 283 (2011) 012016 [arXiv:1012.1558] [INSPIRE].
L. Barclay, The Rich Structure of Gauss-Bonnet Holographic Superconductors, JHEP 10 (2011) 044 [arXiv:1012.3074] [INSPIRE].
R.-G. Cai, Z.-Y. Nie and H.-Q. Zhang, Holographic p-wave superconductors from Gauss-Bonnet gravity, Phys. Rev. D 82 (2010) 066007 [arXiv:1007.3321] [INSPIRE].
R.-G. Cai, Z.-Y. Nie and H.-Q. Zhang, Holographic Phase Transitions of P-wave Superconductors in Gauss-Bonnet Gravity with Back-reaction, Phys. Rev. D 83 (2011) 066013 [arXiv:1012.5559] [INSPIRE].
H.-F. Li, R.-G. Cai and H.-Q. Zhang, Analytical Studies on Holographic Superconductors in Gauss-Bonnet Gravity, JHEP 04 (2011) 028 [arXiv:1103.2833] [INSPIRE].
S. Kanno, A Note on Gauss-Bonnet Holographic Superconductors, Class. Quant. Grav. 28 (2011) 127001 [arXiv:1103.5022] [INSPIRE].
Y. Liu, Q. Pan, B. Wang and R.-G. Cai, Dynamical perturbations and critical phenomena in Gauss-Bonnet-AdS black holes, Phys. Lett. B 693 (2010) 343 [arXiv:1007.2536] [INSPIRE].
S. Gangopadhyay and D. Roychowdhury, Analytic study of Gauss-Bonnet holographic superconductors in Born-Infeld electrodynamics, JHEP 05 (2012) 156 [arXiv:1204.0673] [INSPIRE].
R.-G. Cai and Y.-W. Sun, Shear Viscosity from AdS Born-Infeld Black Holes, JHEP 09 (2008) 115 [arXiv:0807.2377] [INSPIRE].
Q. Pan, J. Jing and B. Wang, Holographic superconductor models with the Maxwell field strength corrections, Phys. Rev. D 84 (2011) 126020 [arXiv:1111.0714] [INSPIRE].
W. Heisenberg and H. Euler, Consequences of Dirac’s theory of positrons, Z. Phys. 98 (1936) 714 [physics/0605038] [INSPIRE].
D.J. Gross and J.H. Sloan, The Quartic Effective Action for the Heterotic String, Nucl. Phys. B 291 (1987) 41 [INSPIRE].
M. Born and L. Infeld, Foundations of the new field theory, Proc. Roy. Soc. Lond. A 144 (1934) 425 [INSPIRE].
G.W. Gibbons and D.A. Rasheed, Electric-magnetic duality rotations in nonlinear electrodynamics, Nucl. Phys. B 454 (1995) 185 [hep-th/9506035] [INSPIRE].
O. Mišković and R. Olea, Conserved charges for black holes in Einstein-Gauss-Bonnet gravity coupled to nonlinear electrodynamics in AdS space, Phys. Rev. D 83 (2011) 024011 [arXiv:1009.5763] [INSPIRE].
B. Hoffmann, Gravitational and Electromagnetic Mass in the Born-Infeld Electrodynamics, Phys. Rev. 47 (1935) 877 [INSPIRE].
H.P. de Oliveira, Nonlinear charged black holes, Class. Quant. Grav. 11 (1994) 1469 [INSPIRE].
H.H. Soleng, Charged black points in general relativity coupled to the logarithmic U(1) gauge theory, Phys. Rev. D 52 (1995) 6178 [hep-th/9509033] [INSPIRE].
Z. Zhao, Q. Pan, S. Chen and J. Jing, Notes on holographic superconductor models with the nonlinear electrodynamics, Nucl. Phys. B 871 (2013) 98 [arXiv:1212.6693] [INSPIRE].
Y. Liu and B. Wang, Perturbations around the AdS Born-Infeld black holes, Phys. Rev. D 85 (2012) 046011 [arXiv:1111.6729] [INSPIRE].
J.-P. Wu, Y. Cao, X.-M. Kuang and W.-J. Li, The 3+1 holographic superconductor with Weyl corrections, Phys. Lett. B 697 (2011) 153 [arXiv:1010.1929] [INSPIRE].
D.-Z. Ma, Y. Cao and J.-P. Wu, The Stückelberg holographic superconductors with Weyl corrections, Phys. Lett. B 704 (2011) 604 [arXiv:1201.2486] [INSPIRE].
J. Jing, Q. Pan and S. Chen, Holographic Superconductors with Power-Maxwell field, JHEP 11 (2011) 045 [arXiv:1106.5181] [INSPIRE].
D. Momeni and M.R. Setare, A note on holographic superconductors with Weyl Corrections, Mod. Phys. Lett. A 26 (2011) 2889 [arXiv:1106.0431] [INSPIRE].
D. Momeni, N. Majd and R. Myrzakulov, p-wave holographic superconductors with Weyl corrections, Europhys. Lett. 97 (2012) 61001 [arXiv:1204.1246] [INSPIRE].
C.O. Lee, The holographic superconductors in higher-dimensional AdS soliton, Eur. Phys. J. C 72 (2012) 2092 [arXiv:1202.5146] [INSPIRE].
R. Banerjee, S. Gangopadhyay, D. Roychowdhury and A. Lala, Holographic s-wave condensate with nonlinear electrodynamics: A nontrivial boundary value problem, Phys. Rev. D 87 (2013) 104001 [arXiv:1208.5902] [INSPIRE].
D. Momeni, M.R. Setare and R. Myrzakulov, Condensation of the scalar field with Stuckelberg and Weyl Corrections in the background of a planar AdS-Schwarzschild black hole, Int. J. Mod. Phys. A 27 (2012) 1250128 [arXiv:1209.3104] [INSPIRE].
J. Jing, Q. Pan and S. Chen, Holographic Superconductor/Insulator Transition with logarithmic electromagnetic field in Gauss-Bonnet gravity, Phys. Lett. B 716 (2012) 385 [arXiv:1209.0893] [INSPIRE].
D. Roychowdhury, Effect of external magnetic field on holographic superconductors in presence of nonlinear corrections, Phys. Rev. D 86 (2012) 106009 [arXiv:1211.0904] [INSPIRE].
D. Roychowdhury, AdS/CFT superconductors with Power Maxwell electrodynamics: reminiscent of the Meissner effect, Phys. Lett. B 718 (2013) 1089 [arXiv:1211.1612] [INSPIRE].
S. Gangopadhyay and D. Roychowdhury, Analytic study of properties of holographic superconductors in Born-Infeld electrodynamics, JHEP 05 (2012) 002 [arXiv:1201.6520] [INSPIRE].
S. Gangopadhyay and D. Roychowdhury, Analytic study of Gauss-Bonnet holographic superconductors in Born-Infeld electrodynamics, JHEP 05 (2012) 156 [arXiv:1204.0673] [INSPIRE].
S. Gangopadhyay and D. Roychowdhury, Analytic study of properties of holographic p-wave superconductors, JHEP 08 (2012) 104 [arXiv:1207.5605] [INSPIRE].
N. Bai, Y.-H. Gao, B.-G. Qi and X.-B. Xu, Holographic insulator/superconductor phase transition in Born-Infeld electrodynamics, arXiv:1212.2721 [INSPIRE].
Z. Zhao, Q. Pan and J. Jing, Holographic insulator/superconductor phase transition with Weyl corrections, Phys. Lett. B 719 (2013) 440 [arXiv:1212.3062] [INSPIRE].
T.K. Dey, Born-Infeld black holes in the presence of a cosmological constant, Phys. Lett. B 595 (2004) 484 [hep-th/0406169] [INSPIRE].
R.-G. Cai, D.-W. Pang and A. Wang, Born-Infeld black holes in (A)dS spaces, Phys. Rev. D 70 (2004) 124034 [hep-th/0410158] [INSPIRE].
T. Nishioka and T. Takayanagi, AdS Bubbles, Entropy and Closed String Tachyons, JHEP 01 (2007) 090 [hep-th/0611035] [INSPIRE].
I.R. Klebanov, D. Kutasov and A. Murugan, Entanglement as a probe of confinement, Nucl. Phys. B 796 (2008) 274 [arXiv:0709.2140] [INSPIRE].
A. Pakman and A. Parnachev, Topological Entanglement Entropy and Holography, JHEP 07 (2008) 097 [arXiv:0805.1891] [INSPIRE].
T. Nishioka, S. Ryu and T. Takayanagi, Holographic Entanglement Entropy: An Overview, J. Phys. A 42 (2009) 504008 [arXiv:0905.0932] [INSPIRE].
L.-Y. Hung, R.C. Myers and M. Smolkin, On Holographic Entanglement Entropy and Higher Curvature Gravity, JHEP 04 (2011) 025 [arXiv:1101.5813] [INSPIRE].
J. de Boer, M. Kulaxizi and A. Parnachev, Holographic Entanglement Entropy in Lovelock Gravities, JHEP 07 (2011) 109 [arXiv:1101.5781] [INSPIRE].
N. Ogawa and T. Takayanagi, Higher Derivative Corrections to Holographic Entanglement Entropy for AdS Solitons, JHEP 10 (2011) 147 [arXiv:1107.4363] [INSPIRE].
T. Albash and C.V. Johnson, Holographic Entanglement Entropy and Renormalization Group Flow, JHEP 02 (2012) 095 [arXiv:1110.1074] [INSPIRE].
R.C. Myers and A. Singh, Comments on Holographic Entanglement Entropy and RG Flows, JHEP 04 (2012) 122 [arXiv:1202.2068] [INSPIRE].
T. Albash and C.V. Johnson, Holographic Studies of Entanglement Entropy in Superconductors, JHEP 05 (2012) 079 [arXiv:1202.2605] [INSPIRE].
X.-M. Kuang, E. Papantonopoulos and B. Wang, Entanglement Entropy as a Probe of the Proximity Effect in Holographic Superconductors, JHEP 05 (2014) 130 [arXiv:1401.5720] [INSPIRE].
W. Yao and J. Jing, Holographic entanglement entropy in metal/superconductor phase transition with Born-Infeld electrodynamics, Nucl. Phys. B 889 (2014) 109 [arXiv:1408.1171] [INSPIRE].
Y. Peng and Y. Liu, A general holographic metal/superconductor phase transition model, JHEP 02 (2015) 082 [arXiv:1410.7234] [INSPIRE].
V.E. Hubeny, M. Rangamani and T. Takayanagi, A Covariant holographic entanglement entropy proposal, JHEP 07 (2007) 062 [arXiv:0705.0016] [INSPIRE].
V. Balasubramanian, A. Bernamonti, J. de Boer, N. Copland, B. Craps, E. Keski-Vakkuri et al., Holographic Thermalization, Phys. Rev. D 84 (2011) 026010 [arXiv:1103.2683] [INSPIRE].
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Liu, Y., Gong, Y. & Wang, B. Non-equilibrium condensation process in holographic superconductor with nonlinear electrodynamics. J. High Energ. Phys. 2016, 116 (2016). https://doi.org/10.1007/JHEP02(2016)116
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DOI: https://doi.org/10.1007/JHEP02(2016)116