Screening of Coulomb interactions in holography

  • E. MauriEmail author
  • H. T. C. Stoof
Open Access
Regular Article - Theoretical Physics


We introduce Coulomb interactions in the holographic description of strongly interacting systems by performing a (current-current) double-trace deformation of the boundary theory. In the theory dual to a Reissner-Nordström background, this deformation leads to gapped plasmon modes in the density-density response, as expected from conventional RPA calculations. We further show that by introducing a (d + 1)-dimensional Coulomb interaction in a boundary theory in d spacetime dimensions, we recover plasmon modes whose dispersion is proportional to \( \sqrt{\left|\mathbf{k}\right|} \), as observed for example in graphene layers. Moreover, motivated by recent experimental results in layered cuprate high-temperature superconductors, we present a toy model for a layered system consisting of an infinite stack of (spatially) two-dimensional layers that are coupled only by the long-range Coulomb interaction. This leads to low-energy ‘acoustic plasmons’. Finally, we compute the optical conductivity of the deformed theory in d = 3 + 1, where a logarithmic correction is present, and we show how this can be related to the conductivity measured in Dirac and Weyl semimetals.


AdS-CFT Correspondence Holography and condensed matter physics (AdS/CMT) 


Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.


  1. [1]
    J.M. Maldacena, The large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    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].ADSCrossRefzbMATHGoogle Scholar
  3. [3]
    E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    C.P. Herzog, Lectures on holographic superfluidity and superconductivity, J. Phys. A 42 (2009) 343001 [arXiv:0904.1975] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  5. [5]
    S.A. Hartnoll, Lectures on holographic methods for condensed matter physics, Class. Quant. Grav. 26 (2009) 224002 [arXiv:0903.3246] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    J. Zaanen, Y. Liu, Y.-W. Sun and K. Schalm, Holographic Duality in Condensed Matter Physics, Cambridge University Press, (2015), [].
  7. [7]
    S.A. Hartnoll, A. Lucas and S. Sachdev, Holographic quantum matter, arXiv:1612.07324 [INSPIRE].
  8. [8]
    H.T.C. Stoof, D.B.M. Dickerscheid and K. Gubbels, Ultracold Quantum Fields (Theoretical and Mathematical Physics), Springer, (2008), [].
  9. [9]
    D. Bohm and D. Pines, A Collective Description of Electron Interactions. 1. Magnetic Interactions, Phys. Rev. 82 (1951) 625 [INSPIRE].
  10. [10]
    D. Pines and D. Bohm, A Collective Description of Electron Interactions: 2. Collective vs Individual Particle Aspects of the Interactions, Phys. Rev. 85 (1952) 338 [INSPIRE].
  11. [11]
    D. Bohm and D. Pines, A Collective Description of Electron Interactions: 3. Coulomb Interactions in a Degenerate Electron Gas, Phys. Rev. 92 (1953) 609 [INSPIRE].
  12. [12]
    D. Pines, A Collective Description of Electron Interactions: 4. Electron Interaction in Metals, Phys. Rev. 92 (1953) 626 [INSPIRE].
  13. [13]
    U. Gran, M. Tornsö and T. Zingg, Holographic Plasmons, JHEP 11 (2018) 176 [arXiv:1712.05672] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    U. Gran, M. Tornsö and T. Zingg, Plasmons in Holographic Graphene, arXiv:1804.02284 [INSPIRE].
  15. [15]
    U. Gran, M. Tornsö and T. Zingg, Exotic Holographic Dispersion, JHEP 02 (2019) 032 [arXiv:1808.05867] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  16. [16]
    U. Gran, M. Tornsö and T. Zingg, Holographic Response of Electron Clouds, JHEP 03 (2019) 019 [arXiv:1810.11416] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  17. [17]
    B. Roy and V. Juričić, Optical conductivity of an interacting Weyl liquid in the collisionless regime, Phys. Rev. B 96 (2017) 155117 [arXiv:1707.08564] [INSPIRE].ADSCrossRefGoogle Scholar
  18. [18]
    A. Lucas and S. Das Sarma, Electronic sound modes and plasmons in hydrodynamic two-dimensional metals, Phys. Rev. B 97 (2018) 115449 [arXiv:1801.01495].ADSCrossRefGoogle Scholar
  19. [19]
    A. Lucas and K.C. Fong, Hydrodynamics of electrons in graphene, J. Phys. Condens. Matter 30 (2018) 053001 [arXiv:1710.08425] [INSPIRE].ADSCrossRefGoogle Scholar
  20. [20]
    M. Hepting et al., Three-dimensional collective charge excitations in electron-doped copper oxide superconductors, Nature 563 (2018) 374.ADSCrossRefGoogle Scholar
  21. [21]
    Y. Ishii and J. Ruvalds, Acoustic plasmons and cuprate superconductivity, Phys. Rev. B 48 (1993) 3455.ADSCrossRefGoogle Scholar
  22. [22]
    V.Z. Kresin and H. Morawitz, Layer plasmons and high-tcsuperconductivity, Phys. Rev. B 37 (1988) 7854.ADSCrossRefGoogle Scholar
  23. [23]
    A. Bill, H. Morawitz and V.Z. Kresin, Electronic collective modes and superconductivity in layered conductors, Phys. Rev. B 68 (2003) 144519.ADSCrossRefGoogle Scholar
  24. [24]
    M. Edalati, J.I. Jottar and R.G. Leigh, Shear Modes, Criticality and Extremal Black Holes, JHEP 04 (2010) 075 [arXiv:1001.0779] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    M. Edalati, J.I. Jottar and R.G. Leigh, Holography and the sound of criticality, JHEP 10 (2010) 058 [arXiv:1005.4075] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  26. [26]
    R.A. Davison and N.K. Kaplis, Bosonic excitations of the AdS 4 Reissner-Nordstrom black hole, JHEP 12 (2011) 037 [arXiv:1111.0660] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  27. [27]
    R.A. Davison and A. Parnachev, Hydrodynamics of cold holographic matter, JHEP 06 (2013) 100 [arXiv:1303.6334] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    J.W. York Jr., Role of conformal three geometry in the dynamics of gravitation, Phys. Rev. Lett. 28 (1972) 1082 [INSPIRE].ADSCrossRefGoogle Scholar
  29. [29]
    G.W. Gibbons and S.W. Hawking, Action Integrals and Partition Functions in Quantum Gravity, Phys. Rev. D 15 (1977) 2752 [INSPIRE].ADSGoogle Scholar
  30. [30]
    S.W. Hawking and G.T. Horowitz, The gravitational Hamiltonian, action, entropy and surface terms, Class. Quant. Grav. 13 (1996) 1487 [gr-qc/9501014] [INSPIRE].
  31. [31]
    S. de Haro, S.N. Solodukhin and K. Skenderis, Holographic reconstruction of space-time and renormalization in the AdS/CFT correspondence, Commun. Math. Phys. 217 (2001) 595 [hep-th/0002230] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  32. [32]
    P.K. Kovtun and A.O. Starinets, Quasinormal modes and holography, Phys. Rev. D 72 (2005) 086009 [hep-th/0506184] [INSPIRE].ADSGoogle Scholar
  33. [33]
    K. Skenderis, Lecture notes on holographic renormalization, Class. Quant. Grav. 19 (2002) 5849 [hep-th/0209067] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  34. [34]
    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].ADSCrossRefzbMATHGoogle Scholar
  35. [35]
    P. Kovtun, Lectures on hydrodynamic fluctuations in relativistic theories, J. Phys. A 45 (2012) 473001 [arXiv:1205.5040] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  36. [36]
    G. Policastro, D.T. Son and A.O. Starinets, From AdS/CFT correspondence to hydrodynamics. 2. Sound waves, JHEP 12 (2002) 054 [hep-th/0210220] [INSPIRE].
  37. [37]
    S.A. Hartnoll, C.P. Herzog and G.T. Horowitz, Building a Holographic Superconductor, Phys. Rev. Lett. 101 (2008) 031601 [arXiv:0803.3295] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  38. [38]
    M.D. Schwartz, Quantum field theory and the standard model, Cambridge University Press, (2014).Google Scholar
  39. [39]
    E. Witten, Multitrace operators, boundary conditions and AdS/CFT correspondence, hep-th/0112258 [INSPIRE].
  40. [40]
    W. Mueck, An improved correspondence formula for AdS/CFT with multitrace operators, Phys. Lett. B 531 (2002) 301 [hep-th/0201100] [INSPIRE].ADSCrossRefGoogle Scholar
  41. [41]
    M. Turlakov and A.J. Leggett, Sum rule analysis of umklapp processes and coulomb energy: Application to cuprate superconductivity, Phys. Rev. B 67 (2003) 094517.ADSCrossRefGoogle Scholar
  42. [42]
    A.L. Fetter, Electrodynamics of a layered electron gas. II. periodic array, Annals Phys. 88 (1974) 1.Google Scholar
  43. [43]
    V.P.J. Jacobs, S.J.G. Vandoren and H.T.C. Stoof, Holographic interaction effects on transport in Dirac semimetals, Phys. Rev. B 90 (2014) 045108 [arXiv:1403.3608] [INSPIRE].ADSCrossRefGoogle Scholar
  44. [44]
    V.P.J. Jacobs, Dirac and Weyl semimetals with holographic interactions, Ph.D. thesis, Utrecht University, The Netherlands, (2015).Google Scholar
  45. [45]
    S.A. Hartnoll, C.P. Herzog and G.T. Horowitz, Holographic Superconductors, JHEP 12 (2008) 015 [arXiv:0810.1563] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  46. [46]
    A. Krikun, A. Romero-Bermúdez, K. Schalm and J. Zaanen, The anomalous attenuation of plasmons in strange metals and holography, arXiv:1812.03968 [INSPIRE].

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.Institute for Theoretical PhysicsUtrecht UniversityUtrechtThe Netherlands

Personalised recommendations