Abstract
In this paper, we construct a new class of topological black hole Lifshitz solutions in the presence of nonlinear exponential electrodynamics for Einstein-dilaton gravity. We show that the reality of Lifshitz supporting Maxwell matter fields exclude the negative horizon curvature solutions except for the asymptotic AdS case. Calculating the conserved and thermodynamical quantities, we obtain a Smarr type formula for the mass and confirm that thermodynamics first law is satisfied on the black hole horizon. Afterward, we study the thermal stability of our solutions and figure out the effects of different parameters on the stability of solutions under thermal perturbations. Next, we apply the gauge/gravity duality in order to calculate the ratio of shear viscosity to entropy for a three-dimensional hydrodynamic system by using the pole method. Furthermore, we study the behavior of holographic conductivity for two-dimensional systems such as graphene. We consider linear Maxwell and nonlinear exponential electrodynamics separately and disclose the effect of nonlinearity on holographic conductivity. We indicate that holographic conductivity vanishes for z > 3 in the case of nonlinear electrodynamics while it does not in the linear Maxwell case. Finally, we solve perturbative additional field equations numerically and plot the behaviors of real and imaginary parts of conductivity for asymptotic AdS and Lifshitz cases. We present experimental results match with our numerical ones.
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R.P. Feynman, QED: the strange theory of light and matter, Princeton University Press, Princeton NJ U.S.A. (1988).
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.A. Hartnoll and P. Kovtun, Hall conductivity from dyonic black holes, Phys. Rev. D 76 (2007) 066001 [arXiv:0704.1160] [INSPIRE].
M. Fujita, W. Li, S. Ryu and T. Takayanagi, Fractional quantum Hall effect via holography: Chern-Simons, edge states and hierarchy, JHEP 06 (2009) 066 [arXiv:0901.0924] [INSPIRE].
S.A. Hartnoll, P.K. Kovtun, M. Muller and S. Sachdev, Theory of the Nernst effect near quantum phase transitions in condensed matter and in dyonic black holes, Phys. Rev. B 76 (2007) 144502 [arXiv:0706.3215] [INSPIRE].
S.A. Hartnoll and C.P. Herzog, Ohm’s law at strong coupling: S duality and the cyclotron resonance, Phys. Rev. D 76 (2007) 106012 [arXiv:0706.3228] [INSPIRE].
S.A. Hartnoll and C.P. Herzog, Impure AdS/CFT correspondence, Phys. Rev. D 77 (2008) 106009 [arXiv:0801.1693] [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.A. Hartnoll, C.P. Herzog and G.T. Horowitz, Holographic superconductors, JHEP 12 (2008) 015 [arXiv:0810.1563] [INSPIRE].
M. Ammon, J. Erdmenger, M. Kaminski and P. Kerner, Flavor superconductivity from gauge/gravity duality, JHEP 10 (2009) 067 [arXiv:0903.1864] [INSPIRE].
G.T. Horowitz and M.M. Roberts, Holographic superconductors with various condensates, Phys. Rev. D 78 (2008) 126008 [arXiv:0810.1077] [INSPIRE].
S.-J. Sin, S.-S. Xu and Y. Zhou, Holographic superconductor for a Lifshitz fixed point, Int. J. Mod. Phys. A 26 (2011) 4617 [arXiv:0909.4857] [INSPIRE].
D.T. Son, Toward an AdS/cold atoms correspondence: a geometric realization of the Schrödinger symmetry, Phys. Rev. D 78 (2008) 046003 [arXiv:0804.3972] [INSPIRE].
A. Adams, K. Balasubramanian and J. McGreevy, Hot spacetimes for cold atoms, JHEP 11 (2008) 059 [arXiv:0807.1111] [INSPIRE].
C.P. Herzog, M. Rangamani and S.F. Ross, Heating up Galilean holography, JHEP 11 (2008) 080 [arXiv:0807.1099] [INSPIRE].
S. Kachru, X. Liu and M. Mulligan, Gravity duals of Lifshitz-like fixed points, Phys. Rev. D 78 (2008) 106005 [arXiv:0808.1725] [INSPIRE].
G. Bertoldi, B.A. Burrington and A. Peet, Black holes in asymptotically Lifshitz spacetimes with arbitrary critical exponent, Phys. Rev. D 80 (2009) 126003 [arXiv:0905.3183] [INSPIRE].
M.H. Dehghani and R.B. Mann, Lovelock-Lifshitz black holes, JHEP 07 (2010) 019 [arXiv:1004.4397] [INSPIRE].
M.H. Dehghani and R.B. Mann, Thermodynamics of Lovelock-Lifshitz black branes, Phys. Rev. D 82 (2010) 064019 [arXiv:1006.3510] [INSPIRE].
M.H. Dehghani and S. Asnafi, Thermodynamics of rotating Lovelock-Lifshitz black branes, Phys. Rev. D 84 (2011) 064038 [arXiv:1107.3354] [INSPIRE].
M.H. Dehghani, C. Shakuri and M.H. Vahidinia, Lifshitz black brane thermodynamics in the presence of a nonlinear electromagnetic field, Phys. Rev. D 87 (2013) 084013 [arXiv:1306.4501] [INSPIRE].
M. Bravo-Gaete and M. Hassaine, Thermodynamics of charged Lifshitz black holes with quadratic corrections, Phys. Rev. D 91 (2015) 064038 [arXiv:1501.03348] [INSPIRE].
R.B. Mann, Lifshitz topological black holes, JHEP 06 (2009) 075 [arXiv:0905.1136] [INSPIRE].
G. Bertoldi, B.A. Burrington and A.W. Peet, Thermal behavior of charged dilatonic black branes in AdS and UV completions of Lifshitz-like geometries, Phys. Rev. D 82 (2010) 106013 [arXiv:1007.1464] [INSPIRE].
J. Tarrio and S. Vandoren, Black holes and black branes in Lifshitz spacetimes, JHEP 09 (2011) 017 [arXiv:1105.6335] [INSPIRE].
M.K. Zangeneh, A. Sheykhi and M.H. Dehghani, Thermodynamics of topological nonlinear charged Lifshitz black holes, Phys. Rev. D 92 (2015) 024050 [arXiv:1506.01784] [INSPIRE].
M.K. Zangeneh, M.H. Dehghani and A. Sheykhi, Thermodynamics of Gauss-Bonnet-dilaton Lifshitz black branes, Phys. Rev. D 92 (2015) 064023 [arXiv:1506.07068] [INSPIRE].
M. Taylor, Non-relativistic holography, arXiv:0812.0530 [INSPIRE].
D. Momeni, R. Myrzakulov, L. Sebastiani and M.R. Setare, Analytical holographic superconductors in AdS N -Lifshitz topological black holes, Int. J. Geom. Meth. Mod. Phys. 12 (2015) 1550015 [arXiv:1210.7965] [INSPIRE].
D. Roychowdhury, Lifshitz holography and the phases of the anisotropic plasma, arXiv:1509.05229 [INSPIRE].
D.-W. Pang, On charged Lifshitz black holes, JHEP 01 (2010) 116 [arXiv:0911.2777] [INSPIRE].
A. Bhattacharyya and D. Roychowdhury, Lifshitz hydrodynamics and new massive gravity, arXiv:1503.03254 [INSPIRE].
S.A. Hartnoll, D.M. Hofman and D. Vegh, Stellar spectroscopy: Fermions and holographic Lifshitz criticality, JHEP 08 (2011) 096 [arXiv:1105.3197] [INSPIRE].
J.P.S. Lemos and D.-W. Pang, Holographic charge transport in Lifshitz black hole backgrounds, JHEP 06 (2011) 122 [arXiv:1106.2291] [INSPIRE].
Y. Bu, Holographic superconductors with z = 2 Lifshitz scaling, Phys. Rev. D 86 (2012) 046007 [arXiv:1211.0037] [INSPIRE].
J.-W. Lu, Y.-B. Wu, P. Qian, Y.-Y. Zhao and X. Zhang, Lifshitz scaling effects on holographic superconductors, Nucl. Phys. B 887 (2014) 112 [arXiv:1311.2699] [INSPIRE].
G. Tallarita, Holographic Lifshitz superconductors with an axion field, Phys. Rev. D 89 (2014) 106005 [arXiv:1402.4691] [INSPIRE].
D. Roychowdhury, Magnetoconductivity in chiral Lifshitz hydrodynamics, JHEP 09 (2015) 145 [arXiv:1508.02002] [INSPIRE].
M. Born and L. Infeld, Foundations of the new field theory, Proc. Roy. Soc. Lond. A 144 (1934) 425 [INSPIRE].
E.S. Fradkin and A.A. Tseytlin, Nonlinear electrodynamics from quantized strings, Phys. Lett. B 163 (1985) 123 [INSPIRE].
E. Bergshoeff, E. Sezgin, C.N. Pope and P.K. Townsend, The Born-Infeld action from conformal invariance of the open superstring, Phys. Lett. B 188 (1987) 70 [INSPIRE].
C.G. Callan Jr., C. Lovelace, C.R. Nappi and S.A. Yost, Loop corrections to superstring equations of motion, Nucl. Phys. B 308 (1988) 221 [INSPIRE].
O.D. Andreev and A.A. Tseytlin, Partition function representation for the open superstring effective action: cancellation of Möbius infinities and derivative corrections to Born-Infeld Lagrangian, Nucl. Phys. B 311 (1988) 205 [INSPIRE].
R.G. Leigh, Dirac-Born-Infeld action from Dirichlet σ-model, Mod. Phys. Lett. A 4 (1989) 2767 [INSPIRE].
S.H. Hendi, Asymptotic charged BTZ black hole solutions, JHEP 03 (2012) 065 [arXiv:1405.4941] [INSPIRE].
S.H. Hendi, Asymptotic Reissner-Nordström black holes, Annals Phys. 333 (2013) 282 [arXiv:1405.5359] [INSPIRE].
S.H. Hendi, Thermodynamic properties of asymptotically Reissner-Nordström black holes, Annals Phys. 346 (2014) 42 [arXiv:1405.6996] [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].
A. Sheykhi and Z. Abdollazadeh, External magnetic field in holographic superconductors with exponential nonlinear electrodynamics, in preparation.
A. Sheykhi and S. Hajkhalili, Dilaton black holes coupled to nonlinear electrodynamic field, Phys. Rev. D 89 (2014) 104019 [arXiv:1504.04009] [INSPIRE].
A. Sheykhi and A. Kazemi, Higher dimensional dilaton black holes in the presence of exponential nonlinear electrodynamics, Phys. Rev. D 90 (2014) 044028 [arXiv:1506.01786] [INSPIRE].
M.K. Zangeneh, A. Sheykhi and M.H. Dehghani, Thermodynamics of topological nonlinear charged Lifshitz black holes, Phys. Rev. D 92 (2015) 024050 [arXiv:1506.01784] [INSPIRE].
M. Abramowitz and I.A. Stegun, Handbook of mathematical functions, Dover, New York U.S.A. (1972).
R.M. Corless, G.H. Gonnet, D.E.G. Hare, D.J. Jeffrey and D.E. Knuth, On the Lambert W function, Adv. Comput. Math. 5 (1996) 329 [INSPIRE].
S.H. Hendi, A. Sheykhi and M.H. Dehghani, Thermodynamics of higher dimensional topological charged AdS black branes in dilaton gravity, Eur. Phys. J. C 70 (2010) 703 [arXiv:1002.0202] [INSPIRE].
J.D. Bekenstein, Black holes and entropy, Phys. Rev. D 7 (1973) 2333 [INSPIRE].
S.W. Hawking, Black hole explosions, Nature 248 (1974) 30 [INSPIRE].
G.W. Gibbons and S.W. Hawking, Cosmological event horizons, thermodynamics and particle creation, Phys. Rev. D 15 (1977) 2738 [INSPIRE].
M.K. Zangeneh, A. Sheykhi and M.H. Dehghani, Thermodynamics of topological nonlinear charged Lifshitz black holes, Phys. Rev. D 92 (2015) 024050 [arXiv:1506.01784] [INSPIRE].
S.W. Hawking, C.J. Hunter and D.N. Page, Nut charge, anti-de Sitter space and entropy, Phys. Rev. D 59 (1999) 044033 [hep-th/9809035] [INSPIRE].
R.B. Mann, Misner string entropy, Phys. Rev. D 60 (1999) 104047 [hep-th/9903229] [INSPIRE].
R.B. Mann, Entropy of rotating Misner string space-times, Phys. Rev. D 61 (2000) 084013 [hep-th/9904148] [INSPIRE].
M. Cvetič and S.S. Gubser, Phases of R charged black holes, spinning branes and strongly coupled gauge theories, JHEP 04 (1999) 024 [hep-th/9902195] [INSPIRE].
M.M. Caldarelli, G. Cognola and D. Klemm, Thermodynamics of Kerr-Newman-AdS black holes and conformal field theories, Class. Quant. Grav. 17 (2000) 399 [hep-th/9908022] [INSPIRE].
S.S. Gubser and I. Mitra, The evolution of unstable black holes in anti-de Sitter space, JHEP 08 (2001) 018 [hep-th/0011127] [INSPIRE].
M.F. Paulos, Transport coefficients, membrane couplings and universality at extremality, JHEP 02 (2010) 067 [arXiv:0910.4602] [INSPIRE].
R.C. Myers, M.F. Paulos and A. Sinha, Holographic studies of quasi-topological gravity, JHEP 08 (2010) 035 [arXiv:1004.2055] [INSPIRE].
Z.Q. Li et al., Dirac charge dynamics in graphene by infrared spectroscopy, Nature Phys. 4 (2008) 532 [arXiv:0807.3780].
S.A. Hartnoll, Lectures on holographic methods for condensed matter physics, Class. Quant. Grav. 26 (2009) 224002 [arXiv:0903.3246] [INSPIRE].
I. Santoso et al., Tunable optical absorption and interactions in graphene via oxygen plasma, Phys. Rev. B 89 (2014) 075134 [arXiv:1307.1358].
D. Tong, Lectures on holographic conductivity, talk presented at Cracow school of theoretical physics, http://www.damtp.cam.ac.uk/user/tong/talks/zakopane.pdf, Poland (2013).
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].
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Zangeneh, M.K., Dehyadegari, A., Sheykhi, A. et al. Thermodynamics and gauge/gravity duality for Lifshitz black holes in the presence of exponential electrodynamics. J. High Energ. Phys. 2016, 37 (2016). https://doi.org/10.1007/JHEP03(2016)037
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DOI: https://doi.org/10.1007/JHEP03(2016)037