A simple holographic superconductor with momentum relaxation

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Regular Article - Theoretical Physics


We study a holographic superconductor model with momentum relaxation due to massless scalar fields linear to spatial coordinates(ψ I  = βδ Ii x i ), where β is the strength of momentum relaxation. In addition to the original superconductor induced by the chemical potential(μ) at β = 0, there exists a new type of superconductor induced by β even at μ = 0. It may imply a new ‘pairing’ mechanism of particles and antiparticles interacting with β, which may be interpreted as ‘impurity’. Two parameters μ and β compete in forming superconducting phase. As a result, the critical temperature behaves differently depending on β/μ. It decreases when β/μ is small and increases when β/μ is large, which is a novel feature compared to other models. After analysing ground states and phase diagrams for various β/μ, we study optical electric(σ), thermoelectric(α), and thermal(\( \overline{\kappa} \)) conductivities. When the system undergoes a phase transition from normal to a superconducting phase, 1 pole appears in the imaginary part of the electric conductivity, implying infinite DC conductivity. If β/μ < 1, at small ω, a two-fluid model with an imaginary 1 pole and the Drude peak works for σ, α, and \( \overline{\kappa} \), but If β/μ > 1 a non-Drude peak replaces the Drude peak. It is consistent with the coherent/incoherent metal transition in its metal phase. The Ferrell-Glover-Tinkham (FGT) sum rule is satisfied for all cases even when μ = 0.


Holography and condensed matter physics (AdS/CMT) Gauge-gravity correspondence 


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  1. [1]
    J. Casalderrey-Solana, H. Liu, D. Mateos, K. Rajagopal and U.A. Wiedemann, Gauge/String Duality, Hot QCD and Heavy Ion Collisions, arXiv:1101.0618 [INSPIRE].
  2. [2]
    S.A. Hartnoll, Lectures on holographic methods for condensed matter physics, Class. Quant. Grav. 26 (2009) 224002 [arXiv:0903.3246] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    C.P. Herzog, Lectures on Holographic Superfluidity and Superconductivity, J. Phys. A 42 (2009) 343001 [arXiv:0904.1975] [INSPIRE].MathSciNetMATHGoogle Scholar
  4. [4]
    N. Iqbal, H. Liu and M. Mezei, Lectures on holographic non-Fermi liquids and quantum phase transitions, arXiv:1110.3814 [INSPIRE].
  5. [5]
    S.A. Hartnoll, C.P. Herzog and G.T. Horowitz, Building a Holographic Superconductor, Phys. Rev. Lett. 101 (2008) 031601 [arXiv:0803.3295] [INSPIRE].ADSCrossRefGoogle Scholar
  6. [6]
    S.A. Hartnoll, C.P. Herzog and G.T. Horowitz, Holographic Superconductors, JHEP 12 (2008) 015 [arXiv:0810.1563] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  7. [7]
    G.T. Horowitz, Introduction to Holographic Superconductors, Lect. Notes Phys. 828 (2011) 313 [arXiv:1002.1722] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  8. [8]
    G.T. Horowitz, J.E. Santos and D. Tong, Optical Conductivity with Holographic Lattices, JHEP 07 (2012) 168 [arXiv:1204.0519] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  9. [9]
    G.T. Horowitz, J.E. Santos and D. Tong, Further Evidence for Lattice-Induced Scaling, JHEP 11 (2012) 102 [arXiv:1209.1098] [INSPIRE].ADSCrossRefGoogle Scholar
  10. [10]
    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].ADSCrossRefGoogle Scholar
  11. [11]
    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].ADSGoogle Scholar
  12. [12]
    A. Donos and J.P. Gauntlett, The thermoelectric properties of inhomogeneous holographic lattices, JHEP 01 (2015) 035 [arXiv:1409.6875] [INSPIRE].ADSCrossRefGoogle Scholar
  13. [13]
    G.T. Horowitz and J.E. Santos, General Relativity and the Cuprates, JHEP 06 (2013) 087 [arXiv:1302.6586] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    D. Vegh, Holography without translational symmetry, arXiv:1301.0537 [INSPIRE].
  15. [15]
    R.A. Davison, Momentum relaxation in holographic massive gravity, Phys. Rev. D 88 (2013) 086003 [arXiv:1306.5792] [INSPIRE].ADSGoogle Scholar
  16. [16]
    M. Blake and D. Tong, Universal Resistivity from Holographic Massive Gravity, Phys. Rev. D 88 (2013) 106004 [arXiv:1308.4970] [INSPIRE].ADSGoogle Scholar
  17. [17]
    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].ADSCrossRefGoogle Scholar
  18. [18]
    A. Donos and J.P. Gauntlett, Holographic Q-lattices, JHEP 04 (2014) 040 [arXiv:1311.3292] [INSPIRE].ADSCrossRefGoogle Scholar
  19. [19]
    A. Donos and J.P. Gauntlett, Novel metals and insulators from holography, JHEP 06 (2014) 007 [arXiv:1401.5077] [INSPIRE].ADSCrossRefGoogle Scholar
  20. [20]
    T. Andrade and B. Withers, A simple holographic model of momentum relaxation, JHEP 05 (2014) 101 [arXiv:1311.5157] [INSPIRE].ADSCrossRefGoogle Scholar
  21. [21]
    B. Goutéraux, Charge transport in holography with momentum dissipation, JHEP 04 (2014) 181 [arXiv:1401.5436] [INSPIRE].ADSCrossRefGoogle Scholar
  22. [22]
    M. Taylor and W. Woodhead, Inhomogeneity simplified, Eur. Phys. J. C 74 (2014) 3176 [arXiv:1406.4870] [INSPIRE].ADSCrossRefGoogle Scholar
  23. [23]
    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].ADSCrossRefGoogle Scholar
  24. [24]
    Y. Bardoux, M.M. Caldarelli and C. Charmousis, Shaping black holes with free fields, JHEP 05 (2012) 054 [arXiv:1202.4458] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  25. [25]
    N. Iizuka and K. Maeda, Study of Anisotropic Black Branes in Asymptotically anti-de Sitter, JHEP 07 (2012) 129 [arXiv:1204.3008] [INSPIRE].ADSCrossRefGoogle Scholar
  26. [26]
    L. Cheng, X.-H. Ge and S.-J. Sin, Anisotropic plasma at finite U(1) chemical potential, JHEP 07 (2014) 083 [arXiv:1404.5027] [INSPIRE].ADSCrossRefGoogle Scholar
  27. [27]
    A. Donos and S.A. Hartnoll, Interaction-driven localization in holography, Nature Phys. 9 (2013) 649 [arXiv:1212.2998] [INSPIRE].ADSCrossRefGoogle Scholar
  28. [28]
    A. Donos, B. Goutéraux and E. Kiritsis, Holographic Metals and Insulators with Helical Symmetry, JHEP 09 (2014) 038 [arXiv:1406.6351] [INSPIRE].ADSCrossRefGoogle Scholar
  29. [29]
    A. Donos, J.P. Gauntlett and C. Pantelidou, Conformal field theories in d = 4 with a helical twist, Phys. Rev. D 91 (2015) 066003 [arXiv:1412.3446] [INSPIRE].ADSMathSciNetGoogle Scholar
  30. [30]
    A. Donos and J.P. Gauntlett, Thermoelectric DC conductivities from black hole horizons, JHEP 11 (2014) 081 [arXiv:1406.4742] [INSPIRE].ADSCrossRefGoogle Scholar
  31. [31]
    H.B. Zeng and J.-P. Wu, Holographic superconductors from the massive gravity, Phys. Rev. D 90 (2014) 046001 [arXiv:1404.5321] [INSPIRE].ADSGoogle Scholar
  32. [32]
    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].ADSCrossRefGoogle Scholar
  33. [33]
    T. Andrade and S.A. Gentle, Relaxed superconductors, arXiv:1412.6521 [INSPIRE].
  34. [34]
    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].ADSGoogle Scholar
  35. [35]
    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].ADSCrossRefGoogle Scholar
  36. [36]
    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].ADSGoogle Scholar
  37. [37]
    I. Amado, M. Kaminski and K. Landsteiner, Hydrodynamics of Holographic Superconductors, JHEP 05 (2009) 021 [arXiv:0903.2209] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  38. [38]
    M. Kaminski, K. Landsteiner, J. Mas, J.P. Shock and J. Tarrio, Holographic Operator Mixing and Quasinormal Modes on the Brane, JHEP 02 (2010) 021 [arXiv:0911.3610] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  39. [39]
    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].ADSCrossRefGoogle Scholar
  40. [40]
    L. Cheng, X.-H. Ge and Z.-Y. Sun, Thermoelectric DC conductivities with momentum dissipation from higher derivative gravity, arXiv:1411.5452 [INSPIRE].
  41. [41]
    R.A. Davison and B. Goutéraux, Momentum dissipation and effective theories of coherent and incoherent transport, JHEP 01 (2015) 039 [arXiv:1411.1062] [INSPIRE].ADSCrossRefGoogle Scholar
  42. [42]
    S.A. Hartnoll, Theory of universal incoherent metallic transport, Nature Phys. 11 (2015) 54 [arXiv:1405.3651] [INSPIRE].ADSCrossRefGoogle Scholar
  43. [43]
    A. Amoretti, A. Braggio, N. Magnoli and D. Musso, Bounds on intrinsic diffusivities in momentum dissipating holography, arXiv:1411.6631 [INSPIRE].
  44. [44]
    F. Denef and S.A. Hartnoll, Landscape of superconducting membranes, Phys. Rev. D 79 (2009) 126008 [arXiv:0901.1160] [INSPIRE].ADSMathSciNetGoogle Scholar
  45. [45]
    G.T. Horowitz and M.M. Roberts, Zero Temperature Limit of Holographic Superconductors, JHEP 11 (2009) 015 [arXiv:0908.3677] [INSPIRE].ADSCrossRefGoogle Scholar
  46. [46]
    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].ADSMathSciNetCrossRefGoogle Scholar
  47. [47]
    K.-Y. Kim, K.K. Kim, and M. Park, work in progress.Google Scholar
  48. [48]
    C.C. Homes, S.V. Dordevic, M. Strongin, D.A. Bonn, R. Liang et al., Universal scaling relation in high-temperature superconductors, Nature 430 (2004) 539 [cond-mat/0404216] [INSPIRE].ADSCrossRefGoogle Scholar
  49. [49]
    J. Erdmenger, P. Kerner and S. Muller, Towards a Holographic Realization of HomesLaw, JHEP 10 (2012) 021 [arXiv:1206.5305] [INSPIRE].ADSCrossRefGoogle Scholar
  50. [50]
    M. Baggioli and O. Pujolàs, Holographic Polarons, the Metal-Insulator Transition and Massive Gravity, arXiv:1411.1003 [INSPIRE].
  51. [51]
    F. Benini, C.P. Herzog and A. Yarom, Holographic Fermi arcs and a d-wave gap, Phys. Lett. B 701 (2011) 626 [arXiv:1006.0731] [INSPIRE].ADSCrossRefGoogle Scholar
  52. [52]
    K.-Y. Kim and M. Taylor, Holographic d-wave superconductors, JHEP 08 (2013) 112 [arXiv:1304.6729] [INSPIRE].ADSCrossRefGoogle Scholar
  53. [53]
    O. Domenech, M. Montull, A. Pomarol, A. Salvio and P.J. Silva, Emergent Gauge Fields in Holographic Superconductors, JHEP 08 (2010) 033 [arXiv:1005.1776] [INSPIRE].ADSCrossRefMATHGoogle Scholar

Copyright information

© The Author(s) 2015

Authors and Affiliations

  1. 1.School of Physics and ChemistryGwangju Institute of Science and TechnologyGwangjuSouth Korea
  2. 2.School of PhysicsKorea Institute for Advanced StudySeoulSouth Korea

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