Abstract
Momentum relaxation can be built into many holographic models without sacrificing homogeneity of the bulk solution. In this paper we study two such models: one in which translational invariance is broken in the dual theory by spatially-dependent sources for massless scalar fields and another that features an additional neutral scalar field. We turn on a charged scalar field in order to explore the condensation of a charged scalar operator in the dual theories. After demonstrating that the relaxed superconductors we construct are thermodynamically relevant, we find that the finite DC electrical conductivity of the normal phase is replaced by a superfluid pole in the broken phase. Moreover, when the normal phase possesses a Drude behaviour at low frequencies, the optical conductivity of the broken phase at low frequencies can be described by a two-fluid model that is a sum of a Drude peak and a superfluid pole, as was found recently for inhomogeneous holographic superconductors. We also study cases in which this low-frequency behavior does not hold. We find that the Drude description is accurate when the retarded current-current correlator has a purely-dissipative pole that is well-separated from the rest of the excitations.
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References
G.T. Horowitz, J.E. Santos and D. Tong, Optical Conductivity with Holographic Lattices, JHEP 07 (2012) 168 [arXiv:1204.0519] [INSPIRE].
G.T. Horowitz, J.E. Santos and D. Tong, Further Evidence for Lattice-Induced Scaling, JHEP 11 (2012) 102 [arXiv:1209.1098] [INSPIRE].
Y. Ling, C. Niu, J.-P. Wu and Z.-Y. Xian, Holographic Lattice in Einstein-Maxwell-Dilaton Gravity, JHEP 11 (2013) 006 [arXiv:1309.4580] [INSPIRE].
A. Donos and J.P. Gauntlett, The thermoelectric properties of inhomogeneous holographic lattices, JHEP 01 (2015) 035 [arXiv:1409.6875] [INSPIRE].
S.A. Hartnoll and D.M. Hofman, Locally Critical Resistivities from Umklapp Scattering, Phys. Rev. Lett. 108 (2012) 241601 [arXiv:1201.3917] [INSPIRE].
P. Chesler, A. Lucas and S. Sachdev, Conformal field theories in a periodic potential: results from holography and field theory, Phys. Rev. D 89 (2014) 026005 [arXiv:1308.0329] [INSPIRE].
A. Karch and A. O’Bannon, Metallic AdS/CFT, JHEP 09 (2007) 024 [arXiv:0705.3870] [INSPIRE].
T. Faulkner, N. Iqbal, H. Liu, J. McGreevy and D. Vegh, From Black Holes to Strange Metals, arXiv:1003.1728 [INSPIRE].
S.A. Hartnoll, J. Polchinski, E. Silverstein and D. Tong, Towards strange metallic holography, JHEP 04 (2010) 120 [arXiv:0912.1061] [INSPIRE].
N. Iizuka et al., Bianchi attractors: a classification of extremal black brane geometries, JHEP 07 (2012) 193 [arXiv:1201.4861] [INSPIRE].
A. Donos and S.A. Hartnoll, Interaction-driven localization in holography, Nature Phys. 9 (2013) 649 [arXiv:1212.2998] [INSPIRE].
T. Andrade and B. Withers, A simple holographic model of momentum relaxation, JHEP 05 (2014) 101 [arXiv:1311.5157] [INSPIRE].
Y. Bardoux, M.M. Caldarelli and C. Charmousis, Shaping black holes with free fields, JHEP 05 (2012) 054 [arXiv:1202.4458] [INSPIRE].
K.-Y. Kim, K.K. Kim, Y. Seo and S.-J. Sin, Coherent/incoherent metal transition in a holographic model, JHEP 12 (2014) 170 [arXiv:1409.8346] [INSPIRE].
S.A. Hartnoll, Theory of universal incoherent metallic transport, Nature Phys. 11 (2015) 54 [arXiv:1405.3651] [INSPIRE].
R.A. Davison and B. Goutéraux, Momentum dissipation and effective theories of coherent and incoherent transport, JHEP 01 (2015) 039 [arXiv:1411.1062] [INSPIRE].
A. Donos and J.P. Gauntlett, Holographic Q-lattices, JHEP 04 (2014) 040 [arXiv:1311.3292] [INSPIRE].
A. Donos and J.P. Gauntlett, Novel metals and insulators from holography, JHEP 06 (2014) 007 [arXiv:1401.5077] [INSPIRE].
B. Goutéraux, Charge transport in holography with momentum dissipation, JHEP 04 (2014) 181 [arXiv:1401.5436] [INSPIRE].
A. Donos and J.P. Gauntlett, Thermoelectric DC conductivities from black hole horizons, JHEP 11 (2014) 081 [arXiv:1406.4742] [INSPIRE].
E. Mefford and G.T. Horowitz, Simple holographic insulator, Phys. Rev. D 90 (2014) 084042 [arXiv:1406.4188] [INSPIRE].
M. Blake and A. Donos, Quantum Critical Transport and the Hall Angle, Phys. Rev. Lett. 114 (2015) 021601 [arXiv:1406.1659] [INSPIRE].
M. Taylor and W. Woodhead, Inhomogeneity simplified, Eur. Phys. J. C 74 (2014) 3176 [arXiv:1406.4870] [INSPIRE].
L. Cheng, X.-H. Ge and Z.-Y. Sun, Thermoelectric DC conductivities with momentum dissipation from higher derivative gravity, JHEP 04 (2015) 135 [arXiv:1411.5452] [INSPIRE].
D. Vegh, Holography without translational symmetry, arXiv:1301.0537 [INSPIRE].
C. de Rham, G. Gabadadze and A.J. Tolley, Resummation of Massive Gravity, Phys. Rev. Lett. 106 (2011) 231101 [arXiv:1011.1232] [INSPIRE].
S. Deser, K. Izumi, Y.C. Ong and A. Waldron, Problems of massive gravities, Mod. Phys. Lett. A 30 (2015) 1540006 [arXiv:1410.2289] [INSPIRE].
M. Blake and D. Tong, Universal Resistivity from Holographic Massive Gravity, Phys. Rev. D 88 (2013) 106004 [arXiv:1308.4970] [INSPIRE].
N. Iqbal and H. Liu, Universality of the hydrodynamic limit in AdS/CFT and the membrane paradigm, Phys. Rev. D 79 (2009) 025023 [arXiv:0809.3808] [INSPIRE].
M. Blake, D. Tong and D. Vegh, Holographic Lattices Give the Graviton an Effective Mass, Phys. Rev. Lett. 112 (2014) 071602 [arXiv:1310.3832] [INSPIRE].
R.A. Davison, Momentum relaxation in holographic massive gravity, Phys. Rev. D 88 (2013) 086003 [arXiv:1306.5792] [INSPIRE].
A. Amoretti, A. Braggio, N. Maggiore, N. Magnoli and D. Musso, Thermo-electric transport in gauge/gravity models with momentum dissipation, JHEP 09 (2014) 160 [arXiv:1406.4134] [INSPIRE].
A. Amoretti, A. Braggio, N. Maggiore, N. Magnoli and D. Musso, Analytic dc thermoelectric conductivities in holography with massive gravitons, Phys. Rev. D 91 (2015) 025002 [arXiv:1407.0306] [INSPIRE].
M. Baggioli and O. Pujolàs, Holographic Polarons, the Metal-Insulator Transition and Massive Gravity, arXiv:1411.1003 [INSPIRE].
A. Adams, D.A. Roberts and O. Saremi, Hawking-Page transition in holographic massive gravity, Phys. Rev. D 91 (2015) 046003 [arXiv:1408.6560] [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].
S.S. Gubser, Breaking an Abelian gauge symmetry near a black hole horizon, Phys. Rev. D 78 (2008) 065034 [arXiv:0801.2977] [INSPIRE].
G.T. Horowitz and J.E. Santos, General Relativity and the Cuprates, JHEP 06 (2013) 087 [arXiv:1302.6586] [INSPIRE].
H.B. Zeng and J.-P. Wu, Holographic superconductors from the massive gravity, Phys. Rev. D 90 (2014) 046001 [arXiv:1404.5321] [INSPIRE].
J.-i. Koga, K. Maeda and K. Tomoda, Holographic superconductor model in a spatially anisotropic background, Phys. Rev. D 89 (2014) 104024 [arXiv:1401.6501] [INSPIRE].
D. Mateos and D. Trancanelli, The anisotropic N = 4 super Yang-Mills plasma and its instabilities, Phys. Rev. Lett. 107 (2011) 101601 [arXiv:1105.3472] [INSPIRE].
D. Mateos and D. Trancanelli, Thermodynamics and Instabilities of a Strongly Coupled Anisotropic Plasma, JHEP 07 (2011) 054 [arXiv:1106.1637] [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].
Y. Ling, P. Liu, C. Niu, J.-P. Wu and Z.-Y. Xian, Holographic Superconductor on Q-lattice, JHEP 02 (2015) 059 [arXiv:1410.6761] [INSPIRE].
F. Aprile and T. Ishii, A Simple Holographic Model of a Charged Lattice, JHEP 10 (2014) 151 [arXiv:1406.7193] [INSPIRE].
R.A. Ferrell and R.E. Glover, Conductivity of Superconducting Films: A Sum Rule, Phys. Rev. 109 (1958) 1398 [INSPIRE].
M. Tinkham and R. Ferrell, Determination of the Superconducting Skin Depth from the Energy Gap and Sum Rule, Phys. Rev. Lett. 2 (1959) 331.
D. Birmingham, I. Sachs and S.N. Solodukhin, Conformal field theory interpretation of black hole quasinormal modes, Phys. Rev. Lett. 88 (2002) 151301 [hep-th/0112055] [INSPIRE].
D.T. Son and A.O. Starinets, Minkowski space correlators in AdS/CFT correspondence: Recipe and applications, JHEP 09 (2002) 042 [hep-th/0205051] [INSPIRE].
M. Henningson and K. Skenderis, The holographic Weyl anomaly, JHEP 07 (1998) 023 [hep-th/9806087] [INSPIRE].
V. Balasubramanian and P. Kraus, A stress tensor for Anti-de Sitter gravity, Commun. Math. Phys. 208 (1999) 413 [hep-th/9902121] [INSPIRE].
J. Sonner and B. Withers, A gravity derivation of the Tisza-Landau Model in AdS/CFT, Phys. Rev. D 82 (2010) 026001 [arXiv:1004.2707] [INSPIRE].
C.C. Homes et al., Universal scaling relation in high-temperature superconductors, Nature 430 (2004) 539 [cond-mat/0404216] [INSPIRE].
C. Homes, S. Dordevic, T. Valla, and M. Strongin, Scaling of the superfluid density in high-temperature superconductors, Phys. Rev B 72 (2005) 134517.
J. Erdmenger, P. Kerner and S. Muller, Towards a Holographic Realization of Homes’ Law, JHEP 10 (2012) 021 [arXiv:1206.5305] [INSPIRE].
G.T. Horowitz and M.M. Roberts, Holographic Superconductors with Various Condensates, Phys. Rev. D 78 (2008) 126008 [arXiv:0810.1077] [INSPIRE].
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Andrade, T., Gentle, S.A. Relaxed superconductors. J. High Energ. Phys. 2015, 140 (2015). https://doi.org/10.1007/JHEP06(2015)140
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DOI: https://doi.org/10.1007/JHEP06(2015)140