Advertisement

Holographic conductivity in disordered systems

  • Shinsei Ryu
  • Tadashi Takayanagi
  • Tomonori Ugajin
Article

Abstract

The main purpose of this paper is to holographically study the behavior of conductivity in 2+1 dimensional disordered systems. We analyze probe D-brane systems in AdS/CFT with random closed string and open string background fields. We give a prescription of calculating the DC conductivity holographically in disordered systems. In particular, we find an analytical formula of the conductivity in the presence of codimension one randomness. We also systematically study the AC conductivity in various probe brane setups without disorder and find analogues of Mott insulators.

Keywords

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

References

  1. [1]
    J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [Adv. Theor. Math. Phys. 2 (1998) 231] [hep-th/9711200] [SPIRES].MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from non-critical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [SPIRES].MathSciNetADSGoogle Scholar
  3. [3]
    E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [SPIRES].MathSciNetMATHGoogle Scholar
  4. [4]
    O. Aharony, S.S. Gubser, J.M. Maldacena, H. Ooguri and Y. Oz, Large-N field theories, string theory and gravity, Phys. Rept. 323 (2000) 183 [hep-th/9905111] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  5. [5]
    S.A. Hartnoll, Lectures on holographic methods for condensed matter physics, Class. Quant. Grav. 26 (2009) 224002 [arXiv:0903.3246] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  6. [6]
    C.P. Herzog, Lectures on holographic superfluidity and superconductivity, J. Phys. A 42 (2009) 343001 [arXiv:0904.1975] [SPIRES].Google Scholar
  7. [7]
    T. Nishioka, S. Ryu and T. Takayanagi, Holographic entanglement entropy: an overview, J. Phys. A 42 (2009) 504008 [arXiv:0905.0932] [SPIRES].MathSciNetGoogle Scholar
  8. [8]
    J. McGreevy, Holographic duality with a view toward many-body physics, Adv. High Energy Phys. 2010 (2010) 723105 [arXiv:0909.0518] [SPIRES].Google Scholar
  9. [9]
    S. Sachdev, Condensed matter and AdS/CFT, arXiv:1002.2947 [SPIRES].
  10. [10]
    S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  11. [11]
    S. Ryu and T. Takayanagi, Aspects of holographic entanglement entropy, JHEP 08 (2006) 045 [hep-th/0605073] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  12. [12]
    S.S. Gubser, Breaking an abelian gauge symmetry near a black hole horizon, Phys. Rev. D 78 (2008) 065034 [arXiv:0801.2977] [SPIRES].ADSGoogle Scholar
  13. [13]
    S.A. Hartnoll, C.P. Herzog and G.T. Horowitz, Building a holographic superconductor, Phys. Rev. Lett. 101 (2008) 031601 [arXiv:0803.3295] [SPIRES].ADSCrossRefGoogle Scholar
  14. [14]
    S.A. Hartnoll, C.P. Herzog and G.T. Horowitz, Holographic superconductors, JHEP 12 (2008) 015 [arXiv:0810.1563] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  15. [15]
    D.T. Son, Toward an AdS/cold atoms correspondence: a geometric realization of the Schroedinger symmetry, Phys. Rev. D 78 (2008) 046003 [arXiv:0804.3972] [SPIRES].MathSciNetADSGoogle Scholar
  16. [16]
    K. Balasubramanian and J. McGreevy, Gravity duals for non-relativistic CFTs, Phys. Rev. Lett. 101 (2008) 061601 [arXiv:0804.4053] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  17. [17]
    S. Kachru, X. Liu and M. Mulligan, Gravity duals of Lifshitz-like fixed points, Phys. Rev. D 78 (2008) 106005 [arXiv:0808.1725] [SPIRES].MathSciNetADSGoogle Scholar
  18. [18]
    K. Goldstein, S. Kachru, S. Prakash and S.P. Trivedi, Holography of charged dilaton black holes, JHEP 08 (2010) 078 [arXiv:0911.3586] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  19. [19]
    K. Goldstein et al., Holography of dyonic dilaton black branes, JHEP 10 (2010) 027 [arXiv:1007.2490] [SPIRES].ADSCrossRefGoogle Scholar
  20. [20]
    H. Liu, J. McGreevy and D. Vegh, Non-Fermi liquids from holography, arXiv:0903.2477 [SPIRES].
  21. [21]
    T. Faulkner, H. Liu, J. McGreevy and D. Vegh, Emergent quantum criticality, Fermi surfaces and AdS 2, arXiv:0907.2694 [SPIRES].
  22. [22]
    T. Faulkner, N. Iqbal, H. Liu, J. McGreevy and D. Vegh, From black holes to strange metals, arXiv:1003.1728 [SPIRES].
  23. [23]
    J. McGreevy, Holographic duality with a view toward many-body physics, Adv. High Energy Phys. 2010 (2010) 723105 [arXiv:0909.0518] [SPIRES].Google Scholar
  24. [24]
    J.L. Davis, P. Kraus and A. Shah, Gravity dual of a quantum Hall plateau transition, JHEP 11 (2008) 020 [arXiv:0809.1876] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  25. [25]
    J. Alanen, E. Keski-Vakkuri, P. Kraus and V. Suur-Uski, AC transport at holographic quantum Hall transitions, JHEP 11 (2009) 014 [arXiv:0905.4538] [SPIRES].ADSCrossRefGoogle Scholar
  26. [26]
    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] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  27. [27]
    Y. Hikida, W. Li and T. Takayanagi, ABJM with flavors and FQHE, JHEP 07 (2009) 065 [arXiv:0903.2194] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  28. [28]
    O. Bergman, N. Jokela, G. Lifschytz and M. Lippert, Quantum Hall effect in a holographic model, JHEP 10 (2010) 063 [arXiv:1003.4965] [SPIRES].ADSCrossRefGoogle Scholar
  29. [29]
    S. Ryu and T. Takayanagi, Topological insulators and superconductors from D-branes, Phys. Lett. B 693 (2010) 175 [arXiv:1001.0763] [SPIRES].MathSciNetADSGoogle Scholar
  30. [30]
    S. Ryu and T. Takayanagi, Topological insulators and superconductors from string theory, Phys. Rev. D 82 (2010) 086014 [arXiv:1007.4234] [SPIRES].ADSGoogle Scholar
  31. [31]
    C. Hoyos-Badajoz, K. Jensen and A. Karch, A holographic fractional topological insulator, Phys. Rev. D 82 (2010) 086001 [arXiv:1007.3253] [SPIRES].ADSGoogle Scholar
  32. [32]
    A. Karch, J. Maciejko and T. Takayanagi, Holographic fractional topological insulators in 2 + 1 and 1 + 1 dimensions, Phys. Rev. D 82 (2010) 126003 [arXiv:1009.2991] [SPIRES].ADSGoogle Scholar
  33. [33]
    P.W. Anderson, Absence of diffusion in certain random lattices, Phys. Rev. 109 (1958) 1492.ADSCrossRefGoogle Scholar
  34. [34]
    E. Abrahams et al., Scaling theory of localization: absence of quantum diffusion in two dimensions, Phys. Rev. Lett. 42 (1979) 673. ADSCrossRefGoogle Scholar
  35. [35]
    P.A. Lee and T.V. Ramakrishnan, Disordered electronic systems, Rev. Mod. Phys. 57 (1985) 287 [SPIRES].ADSCrossRefGoogle Scholar
  36. [36]
    D. Belitz and T.R. Kirkpatrick, The Anderson-Mott transition, Rev. Mod. Phys. 66 (1994) 261 [SPIRES].ADSCrossRefGoogle Scholar
  37. [37]
    F. Evers and A.D. Mirlin, Anderson transitions, Rev. Mod. Phys. 80 (2008) 1355 [SPIRES].ADSCrossRefGoogle Scholar
  38. [38]
    S.V. Kravchenko and M.P. Sarachik, Metal-insulator transition in two-dimensional electron systems, Rep. Prog. Phys. 67 (2004) 1.ADSCrossRefGoogle Scholar
  39. [39]
    A. Punnoose and A.M. Finkel’stein, Metal-insulator transition in disordered two-dimensional electron systems, Science 310 (2005) 289.ADSCrossRefGoogle Scholar
  40. [40]
    S. Das Sarma et al., Electronic transport in two dimensional graphene, arXiv:1003.4731.
  41. [41]
    S.A. Hartnoll and C.P. Herzog, Impure AdS/CFT, Phys. Rev. D 77 (2008) 106009 [arXiv:0801.1693] [SPIRES].MathSciNetADSGoogle Scholar
  42. [42]
    M. Fujita, Y. Hikida, S. Ryu and T. Takayanagi, Disordered systems and the replica method in AdS/CFT, JHEP 12 (2008) 065 [arXiv:0810.5394] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  43. [43]
    S. Kachru, A. Karch and S. Yaida, Holographic lattices, dimers and glasses, Phys. Rev. D 81 (2010) 026007 [arXiv:0909.2639] [SPIRES].MathSciNetADSGoogle Scholar
  44. [44]
    S. Kachru, A. Karch and S. Yaida, Adventures in holographic dimer models, New J. Phys. 13 (2011) 035004 [arXiv:1009.3268] [SPIRES].ADSCrossRefGoogle Scholar
  45. [45]
    M.S. Foster, S. Ryu and A.W.W. Ludwig, Termination of typical wavefunction multifractal spectra at the Anderson metal-insulator transition: field theory description using the functional renormalization group, Phys. Rev. B 80 (2009) 075101. ADSGoogle Scholar
  46. [46]
    T. Vojta Atypical is normal at the metal-insulator transition, Physics 2 (2009) 66.CrossRefGoogle Scholar
  47. [47]
    A. Adams and S. Yaida, Disordered holographic systems I: functional renormalization, arXiv:1102.2892 [SPIRES].
  48. [48]
    L.-Y. Hung and Y. Shang, On 1-loop diagrams in AdS space, Phys. Rev. D 83 (2011) 024029 [arXiv:1007.2653] [SPIRES].ADSGoogle Scholar
  49. [49]
    A. Karch and E. Katz, Adding flavor to AdS/CFT, JHEP 06 (2002) 043 [hep-th/0205236] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  50. [50]
    A. Karch and A. O’Bannon, Metallic AdS/CFT, JHEP 09 (2007) 024 [arXiv:0705.3870] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  51. [51]
    A. Karch and L. Randall, Localized gravity in string theory, Phys. Rev. Lett. 87 (2001) 061601 [hep-th/0105108] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  52. [52]
    A. Karch and L. Randall, Open and closed string interpretation of SUSY CFT’s on branes with boundaries, JHEP 06 (2001) 063 [hep-th/0105132] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  53. [53]
    R.C. Myers and M.C. Wapler, Transport properties of holographic defects, JHEP 12 (2008) 115 [arXiv:0811.0480] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  54. [54]
    S.A. Hartnoll, J. Polchinski, E. Silverstein and D. Tong, Towards strange metallic holography, JHEP 04 (2010) 120 [arXiv:0912.1061] [SPIRES].ADSCrossRefGoogle Scholar
  55. [55]
    C.-M. Chen and D.-W. Pang, Holography of charged dilaton black holes in general dimensions, JHEP 06 (2010) 093 [arXiv:1003.5064] [SPIRES].ADSCrossRefGoogle Scholar
  56. [56]
    S.R. Das, T. Nishioka and T. Takayanagi, Probe branes, time-dependent couplings and thermalization in AdS/CFT, JHEP 07 (2010) 071 [arXiv:1005.3348] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  57. [57]
    C. Charmousis, B. Gouteraux, B.S. Kim, E. Kiritsis and R. Meyer, Effective holographic theories for low-temperature condensed matter systems, JHEP 11 (2010) 151 [arXiv:1005.4690] [SPIRES].ADSCrossRefGoogle Scholar
  58. [58]
    C. Hoyos-Badajoz, A. O’Bannon and J.M.S. Wu, Zero sound in strange metallic holography, JHEP 09 (2010) 086 [arXiv:1007.0590] [SPIRES].ADSCrossRefGoogle Scholar
  59. [59]
    M. Imada, A. Fujimori and Y. Tokura, Metal-insulator transitions, Rev. Mod. Phys. 70 (1998) 1039 [SPIRES].ADSCrossRefGoogle Scholar
  60. [60]
    U. Gürsoy, E. Kiritsis, L. Mazzanti and F. Nitti, Holography and thermodynamics of 5D dilaton-gravity, JHEP 05 (2009) 033 [arXiv:0812.0792] [SPIRES].CrossRefGoogle Scholar
  61. [61]
    T. Azeyanagi, W. Li and T. Takayanagi, On string theory duals of Lifshitz-like fixed points, JHEP 06 (2009) 084 [arXiv:0905.0688] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  62. [62]
    D.T. Son and A.O. Starinets, Minkowski-space correlators in AdS/CFT correspondence: recipe and applications, JHEP 09 (2002) 042 [hep-th/0205051] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  63. [63]
    M. Ammon, J. Erdmenger, M. Kaminski and P. Kerner, Superconductivity from gauge/gravity duality with flavor, Phys. Lett. B 680 (2009) 516 [arXiv:0810.2316] [SPIRES].ADSGoogle Scholar
  64. [64]
    T. Nishioka, S. Ryu and T. Takayanagi, Holographic superconductor/insulator transition at zero temperature, JHEP 03 (2010) 131 [arXiv:0911.0962] [SPIRES].ADSCrossRefGoogle Scholar
  65. [65]
    K. Balasubramanian and J. McGreevy, The particle number in galilean holography, JHEP 01 (2011) 137 [arXiv:1007.2184] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  66. [66]
    M. Edalati, R.G. Leigh and P.W. Phillips, Dynamically generated Mott gap from holography, Phys. Rev. Lett. 106 (2011) 091602 [arXiv:1010.3238] [SPIRES].ADSCrossRefGoogle Scholar
  67. [67]
    E. Witten, Anti-de Sitter space, thermal phase transition and confinement in gauge theories, Adv. Theor. Math. Phys. 2 (1998) 505 [hep-th/9803131] [SPIRES].MathSciNetMATHGoogle Scholar
  68. [68]
    G.T. Horowitz and R.C. Myers, The AdS/CFT correspondence and a new positive energy conjecture for general relativity, Phys. Rev. D 59 (1998) 026005 [hep-th/9808079] [SPIRES].MathSciNetADSGoogle Scholar
  69. [69]
    H.U. Baranger and A.D. Stone, Electrical linear-response theory in an arbitrary magnetic field: A new Fermi-surface formation, Phys. Rev. B 40 (1989) 8169. ADSGoogle Scholar
  70. [70]
    S. Xiong, N. Read and A.D. Stone, Mesoscopic conductance and its fluctuations at a non-zero Hall angle, Phys. Rev. B 56 (1997) 3982 [cond-mat/9701077].ADSGoogle Scholar
  71. [71]
    P.M. Ostrovsky, I.V. Gornyi and A.D. Mirlin, Quantum criticality and minimal conductivity in graphene with long-range disorder, Phys. Rev. Lett. 98 (2007) 256801 [cond-mat/0702115].ADSCrossRefGoogle Scholar
  72. [72]
    S. Ryu et al., Z2 topological term, the global anomaly, and the two-dimensional symplectic symmetry class of Anderson localization, Phys. Rev. Lett. 99 (2007) 116601.ADSCrossRefGoogle Scholar
  73. [73]
    J.H. Bardarson et al., Demonstration of one-parameter scaling at the Dirac point in graphene, Phys. Rev. Lett. 99 (2007) 106801 [arXiv:0705.0886].ADSCrossRefGoogle Scholar
  74. [74]
    K. Nomura, M. Koshino and S. Ryu, Topological delocalization of two-dimensional massless Dirac fermions, Phys. Rev. Lett. 99 (2007) 146806 [arXiv:0705.1607].ADSCrossRefGoogle Scholar
  75. [75]
    A.W.W. Ludwig, M.P.A. Fisher, R. Shankar and G. Grinstein, Integer quantum Hall transition: an alternative approach and exact results, Phys. Rev. B 50 (1994) 7526. ADSGoogle Scholar
  76. [76]
    S. Cho and M.P.A. Fisher, Conductance fluctuations at the integer quantum Hall plateau transition, Phys. Rev. B 55 (1997) 1673. Google Scholar
  77. [77]
    K. Nomura et al., Quantum Hall effect of massless Dirac fermions in a vanishing magnetic field, Phys. Rev. Lett. 100 (2008) 246806 [arXiv:0801.3121].ADSCrossRefGoogle Scholar
  78. [78]
    O. Aharony, S. Minwalla and T. Wiseman, Plasma-balls in large-N gauge theories and localized black holes, Class. Quant. Grav. 23 (2006) 2171 [hep-th/0507219] [SPIRES].MathSciNetMATHCrossRefGoogle Scholar
  79. [79]
    C.L. Kane and M.P.A. Fisher, Transport in a one-channel Luttinger liquid, Phys. Rev. Lett. 68 (1992) 1220. ADSCrossRefGoogle Scholar
  80. [80]
    A. Furusaki and N. Nagaosa, Single-barrier problem and Anderson localization in a one-dimensional interacting electron system, Phys. Rev. B 47 (1993) 4631. ADSGoogle Scholar
  81. [81]
    D.R. Gulotta, C.P. Herzog and M. Kaminski, Sum rules from an extra dimension, JHEP 01 (2011) 148 [arXiv:1010.4806] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  82. [82]
    G.D. Mahan, Many-particle physics, 3rd edition, Kluwer Academic/Plenum Publishers, U.S.A. (2000).Google Scholar

Copyright information

© SISSA, Trieste, Italy 2011

Authors and Affiliations

  • Shinsei Ryu
    • 1
  • Tadashi Takayanagi
    • 2
  • Tomonori Ugajin
    • 2
  1. 1.Department of PhysicsUniversity of CaliforniaBerkeleyU.S.A.
  2. 2.Institute for the Physics and Mathematics of the Universe (IPMU)University of TokyoKashiwa, ChibaJapan

Personalised recommendations