Skip to main content
SpringerLink
Log in
Book cover

From Gravity to Thermal Gauge Theories: The AdS/CFT Correspondence pp 313–347Cite as

Introduction to Holographic Superconductors

  • Chapter
  • First Online:

Part of the book series: Lecture Notes in Physics ((LNP,volume 828))

Abstract

These lectures give an introduction to the theory of holographic superconductors. These are superconductors that have a dual gravitational description using gauge/gravity duality. After introducing a suitable gravitational theory, we discuss its properties in various regimes: the probe limit, the effects of backreaction, the zero temperature limit, and the addition of magnetic fields. Using the gauge/gravity dictionary, these properties reproduce many of the standard features of superconductors. Some familiarity with gauge/gravity duality is assumed. A list of open problems is included at the end.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Notes

  1. 1.

    A good general reference is [63].

  2. 2.

    Discussion at the KITP program on Quantum Criticality and the AdS/CFT Correspondence, July 2009.

  3. 3.

    I will assume that the reader is familiar with the basics of gauge/gravity duality. If not, see other contributions in this book.

  4. 4.

    This means that one imagines that the dual action includes terms like \(|\nabla_i - ie A_i)\varphi|^2\) with very small charge \(e\).

  5. 5.

    I thank Karl Landsteiner for suggesting this.

  6. 6.

    We will work in the classical limit, but the correspondence applies to the full quantum theory.

  7. 7.

    This normalization of \(O\) differs from that of [36] by a factor of \(\sqrt{2}\).

  8. 8.

    A few caveats should be noted: If one adds scalar interactions in the bulk by introducing a more general potential \(V(\Uppsi)\), the gap in the low temperature optical conductivity can become much less pronounced, so that this ratio becomes ill defined [28, 29]. We will see in the next section that it also becomes ill defined at small \(q\). Even when it is well defined, it is modified by higher order curvature corrections in bulk [24].

  9. 9.

    We will set \(L=1\) for the rest of our discussion.

  10. 10.

    I thank Hong Liu for this comment.

  11. 11.

    See [2,3] for another approach.

  12. 12.

    In general, there is an additional neutral scalar, but for purely electrically charged black holes, this field can be set to zero.

  13. 13.

    One cannot compute the energy gap directly. Indeed, as we have seen, there probably is not a strict gap. The interpretation as a P-wave superconductor comes from the fact that there is a vector order parameter and the conductivity is strongly anisotropic in a manor consistent with P-wave nodes on the Fermi surface.

  14. 14.

    This is not an issue for the \(3+1\) dimensional superconductor, which can be holographically described by the bulk gravitational theory (10.2) in one higher dimension.

  15. 15.

    I thank Sean Hartnoll for suggesting some of these problems.

References

  1. Albash, T., Johnson, C.V.: A holographic superconductor in an external magnetic field. JHEP 0809, 121 (2008)

    Google Scholar 

  2. Albash, T., Johnson, C.V.: Phases of holographic superconductors in an external magnetic field. Phys. Rev. D arXiv:0906.0519 [hep-th]

    Google Scholar 

  3. Albash, T., Johnson, C.V.: Vortex and droplet engineering in holographic superconductors. Phys. Rev. D 80, 126009 (2009)

    Google Scholar 

  4. Amado, I., Kaminski, M., Landsteiner, K.: Hydrodynamics of holographic superconductors. JHEP 0905, 021 (2009)

    Google Scholar 

  5. Ammon, M., Erdmenger, J., Kaminski, M., Kerner, P.: Superconductivity from gauge/gravity duality with flavor. Phys. Lett. B 680, 516 (2009)

    Google Scholar 

  6. Ammon, M., Erdmenger, J., Kaminski, M., Kerner, P.: Flavor Super- conductivity from gauge/gravity duality. JHEP 0910, 067 (2009)

    Google Scholar 

  7. Bardeen, J., Cooper, L.N., Schrieffer, J.R.: Theory of superconductivity. Phys. Rev. 108, 1175 (1957)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  8. Basu, P., Mukherjee, A., Shieh, H.H.: Supercurrent: vector hair for an AdS black hole. Phys. Rev. D 79, 045010 (2009)

    Google Scholar 

  9. Bednorz, J.G., Muller, K.A.: Possible high T c superconductivity in the Ba-La-Cu-O system. Z. Phys. B 64, 189 (1986)

    Article  ADS  Google Scholar 

  10. Bekenstein, J.D.: Black hole hair: twenty-five years after. (1996) [arXiv:gr-qc/9605059]

    Google Scholar 

  11. Breitenlohner, P., Freedman D.Z.: Stability in gauged extended supergravity. Ann. Phys. 144, 249 (1982)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  12. Chen, T.Y., Tesanovic, Z., Liu, R.H., Chen, X.H., Chien, C.L.: A BCS-like gap in the superconductor SmFeAsOF. Nature 453, 1224 (2008)

    Article  ADS  Google Scholar 

  13. Chen, J.W., Kao, Y.J., Wen, W.Y.: Peak-dip-hump from holographic superconductivity. arXiv:0911.2821 [hep-th]

    Google Scholar 

  14. Denef, F., Hartnoll, S.A.: Landscape of superconducting membranes. Phys. Rev. D 79, 126008 (2009)

    Google Scholar 

  15. D’Hoker, E., Kraus, P.: Magnetic brane solutions in AdS. JHEP 0910, 088 (2009)

    Google Scholar 

  16. D’Hoker, E., Kraus, P.: Charged magnetic brane solutions in AdS 5 and the fate of the third law of thermodynamics. (2009) arXiv:0911.4518 [hep-th]

    Google Scholar 

  17. Faulkner, T., Horowitz, G.T., McGreevy, J.,Roberts, M.M., Vegh, D.: Photoemission ’experiments’ on holographic superconductors. arXiv:0911.3402 [hep-th]

    Google Scholar 

  18. Franco, S., Garcia-Garcia, A., Rodriguez-Gomez, D.: A general class of holographic superconductors. arXiv:0906.1214 [hep-th]

    Google Scholar 

  19. Franco, S., Garcia-Garcia, A., Rodriguez-Gomez, D.: A holographic approach to phase transitions. arXiv:0911.1354 [hep-th]

    Google Scholar 

  20. Gauntlett, J.P., Sonner, J., Wiseman, T.: Holographic superconductivity in M-Theory. Phys. Rev. Lett. 103, 151601 (2009)

    Google Scholar 

  21. Gauntlett, J.P., Sonner, J., Wiseman, T.: Quantum criticality and holographic superconductors in M-theory. arXiv:0912.0512 [hep-th]

    Google Scholar 

  22. Ginzburg, V.L., Landau, L.D.: On the theory of superconductivity. Zh. Eksp. Teor. Fiz. 20, 1064 (1950)

    Google Scholar 

  23. Gomes, K.K., Pasupathy, A.N., Pushp, A., Ono, S., Ando, Y., Yazdani, A..: Visualizing pair formation on the atomic scale in the high-Tc superconductor Bi 2 Sr 2 CaCu 2 O 8+δ. Nature 447, 569 (2007)

    Article  ADS  Google Scholar 

  24. Gregory, R., Kanno, S., Soda, J.: Holographic superconductors with higher curvature corrections. JHEP 0910, 010 (2009)

    Google Scholar 

  25. Gubser, S.S.: Breaking an Abelian gauge symmetry near a black hole horizon. Phys. Rev. D 78, 065034 (2008)

    Google Scholar 

  26. Gubser, S.S., Klebanov, I.R., Polyakov, A.M.: Gauge theory correlators from non-critical string theory. Phys. Lett. B 428, 105 (1998)

    Google Scholar 

  27. Gubser, S.S., Pufu, S.S.: The gravity dual of a p-wave superconductor. JHEP 0811, 033 (2008)

    Google Scholar 

  28. Gubser, S.S., Nellore, A.: Low-temperature behavior of the Abelian Higgs model in anti-de Sitter space. JHEP 0904, 008 (2009)

    Google Scholar 

  29. Gubser, S.S., Nellore, A.: Ground states of holographic superconductors. Phys. Rev. D 80, 105007 (2009)

    Google Scholar 

  30. Gubser, S.S., Pufu, S.S., Rocha, F.D.: Quantum critical superconductors in string theory and M-theory. Phys. Lett. B 683, 201 (2010)

    Google Scholar 

  31. Gubser, S.S., Rocha, F.D., Talavera, P.: Normalizable fermion modes in a holographic superconductor. arXiv:0911.3632 [hep-th]

    Google Scholar 

  32. Gubser, S.S., Herzog, C.P., Pufu, S.S., Tesileanu, T.: Superconductors from superstrings. Phys. Rev. Lett. 103, 141601 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  33. Hartnoll, S.A.: Lectures on holographic methods for condensed matter physics. Class. Quant. Grav. 26, 224002 (2009)

    Google Scholar 

  34. Hartnoll, S.A.: Quantum critical dynamics from black holes. arXiv:0909.3553 [cond-mat.str-el]

    Google Scholar 

  35. Hartnoll, S.A., Herzog, C.P.: Ohm’s Law at strong coupling: S duality and the cyclotron resonance. Phys. Rev. D 76, 106012 (2007)

    Google Scholar 

  36. Hartnoll, S.A., Herzog, C.P., Horowitz, G.T.: Building a holographic superconductor. Phys. Rev. Lett. 101, 031601 (2008)

    Google Scholar 

  37. Hartnoll, S.A., Herzog, C.P., Horowitz, G.T.: Holographic superconductors. JHEP 0812, 015 (2008)

    Google Scholar 

  38. Hertog, T.: Towards a novel no-hair theorem for black holes. Phys. Rev. D 74, 084008 (2006)

    Google Scholar 

  39. Herzog C.P.: Lectures on holographic superfluidity and superconductivity. J. Phys. A 42, 343001 (2009)

    Google Scholar 

  40. Herzog, C.P., Kovtun, P., Sachdev, S., Son, D.T.: Quantum critical transport, duality, and M-theory. Phys. Rev. D 75, 085020 (2007)

    Google Scholar 

  41. Herzog, C.P., Kovtun, P.K., Son, D.T.: Holographic model of superfluidity. Phys. Rev. D 79, 066002 (2009)

    Google Scholar 

  42. Heusler, M.: No-hair theorems and black holes with hair. Helv. Phys. Acta 69, 501 (1996)

    Google Scholar 

  43. Homes, C.C. et al.: Universal scaling relation in high-temperature superconductors. Nature 430 (2004) 539

    Google Scholar 

  44. Horowitz, G.T., Roberts, M.M.: Holographic superconductors with various condensates. Phys. Rev. D 78, 126008 (2008)

    Google Scholar 

  45. Horowitz, G.T., Roberts, M.M.: Zero temperature limit of holographic superconductors. JHEP 0911, 015 (2009)

    Google Scholar 

  46. Kachru, S., Karch, A., Yaida, S.: Holographic lattices, dimers, and glasses. Phys. Rev. D 81, 026007 (2010)

    Google Scholar 

  47. Kamihara, Y., Watanabe, T., Hirano, M., Hosono, H. : J. Am. Chem. Soc. 130, 3296 (2008)

    Article  Google Scholar 

  48. Kim,Y., Ko, Y., Sin, S.J.: Density driven symmetry breaking and butterfly effect in holographic superconductors. Phys. Rev. D 80, 126017 (2009)

    Google Scholar 

  49. Klebanov, I.R., Witten, E.: AdS/CFT correspondence and symmetry breaking. Nucl. Phys. B 556, 89 (1999)

    Google Scholar 

  50. Koutsoumbas, G., Papantonopoulos, E., Siopsis, G.: Exact gravity dual of a gapless superconductor. JHEP 0907, 026 (2009)

    Google Scholar 

  51. London, F., London, H.: Proc. Roy. Soc. (London) A149, 71 (1935)

    Article  ADS  MATH  Google Scholar 

  52. Maeda, K., Okamura, T.: Characteristic length of an AdS/CFT superconductor. Phys. Rev. D 78, 106006 (2008)

    Google Scholar 

  53. Maeda, K., Natsuume, M., Okamura, T.: Vortex lattice for a holographic superconductor. Phys. Rev. D 81, 026002 (2010)

    Google Scholar 

  54. Maldacena, J.M.: The large N limit of superconformal field theories and supergravity. Adv. Theor. Math. Phys. 2, 231 (1998) [Int. J. Theor. Phys. 38, 1113 (1999)]

    Google Scholar 

  55. McGreevy, J.: Holographic duality with a view toward many-body physics. arXiv:0909.0518 [hep-th]

    Google Scholar 

  56. Montull, M., Pomarol, A., Silva, P.J.: The holographic superconductor vortex. Phys. Rev. Lett. 103, 091601 (2009)

    Google Scholar 

  57. Nakamura, S., Ooguri, H., Park, C.S.: Gravity dual of spatially modulated phase. arXiv:0911.0679 [hep-th]

    Google Scholar 

  58. Peeters, K., Powell, J., Zamaklar, M.: Exploring colourful holographic superconductors. JHEP 0909, 101 (2009)

    Google Scholar 

  59. Polchinski, J.: KITP talk during the workshop on quantum criticality and AdS/CFT correspondence, July 21, 2009

    Google Scholar 

  60. Roberts, M.M., Hartnoll, S.A.: Pseudogap and time reversal breaking in a holographic superconductor. JHEP 0808, 035 (2008)

    Google Scholar 

  61. Son, D.T., Starinets, A.O.: Minkowski-space correlators in AdS/CFT correspondence: recipe and applications. JHEP 0209, 042 (2002)

    Google Scholar 

  62. Sonner, J.: A Rotating holographic superconductor. Phys. Rev. D 80, 084031 (2009)

    Google Scholar 

  63. Tinkham, M.: Introduction to Superconductivity, 2nd (edn.) Dover, New York (1996)

    Google Scholar 

  64. Vinokur, V., Baturina, T., Fistul, M., Mironov, A., Baklanov, M., Strunk, C.: Superinsulator and quantum synchronization. Nature 452, 613 (2008)

    Article  ADS  Google Scholar 

  65. Volkov, M.S., Galtsov, D.V.: NonAbelian Einstein Yang-Mills black holes. JETP Lett. 50, 346 (1989) [Pisma Zh. Eksp.Teor. Fiz. 50, 312 (1989)]

    Google Scholar 

  66. Weinberg, S.: Superconductivity for particular theorists. Prog. Theor. Phys. Suppl. 86, 43 (1986)

    Article  ADS  Google Scholar 

  67. Witten, E.: Anti-de Sitter space and holography. Adv. Theor. Math. Phys. 2, 253 (1998)

    Google Scholar 

Download references

Acknowledgement

It is a pleasure to thank my collaborators, Sean Hartnoll, Chris Herzog, and Matt Roberts for teaching me many of the results described here. I also thank Hartnoll and Roberts for comments on these lecture notes. Finally, I thank the organizers and participants of the 5th Aegean Summer School, “From Gravity to Thermal Gauge Theories: the AdS/CFT Correspondence” for stimulating discussions. This work is supported in part by NSF grant number PHY-0855415.

Author information

Authors and Affiliations

  1. Physics Department, UCSB, Santa Barbara, CA, 93106, USA

    Gary T. Horowitz

Authors
  1. Gary T. Horowitz
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gary T. Horowitz .

Editor information

Editors and Affiliations

  1. Dept.Physics, National Technical University, Zografou Campus, Athens, 157 80, Greece

    Eleftherios Papantonopoulos

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Horowitz, G.T. (2011). Introduction to Holographic Superconductors. In: Papantonopoulos, E. (eds) From Gravity to Thermal Gauge Theories: The AdS/CFT Correspondence. Lecture Notes in Physics, vol 828. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04864-7_10

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-04864-7_10

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-04863-0

  • Online ISBN: 978-3-642-04864-7

  • eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)

Publish with us

Policies and ethics

Search

Navigation