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On the AdS/BCFT approach to quantum Hall systems

Abstract

In this paper we study a simple gravity model dual to a 2 + 1-dimensional system with a boundary at finite charge density and temperature. In our naive AdS/BCF T extension of a well known AdS/CFT system a non-zero charge density must be supported by a magnetic field. As a result, the Hall conductivity is a constant inversely proportional to the coefficients of pertinent topological terms. Since the direct conductivity vanishes, such behaviors resemble that of a quantum Hall system with Fermi energy in the gap between the Landau levels. We further analyze the properties stemming from our holographic approach to a quantum Hall system. We find that at low temperatures the thermal and electric conductivities are related through the Wiedemann-Franz law, so that every charge conductance mode carries precisely one quantum of the heat conductance. From the computation of the edge currents we learn that the naive holographic model is dual to a gapless system if tensionless RS branes are used in the AdS/BCFT construction. To reconcile this result with the expected quantum Hall behavior we conclude that gravity solutions with tensionless RS branes must be unstable, calling for a search of more general solutions. We briefly discuss the expected features of more realistic holographic setups.

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References

  1. T. Takayanagi, Holographic dual of BCFT, Phys. Rev. Lett. 107 (2011) 101602 [arXiv:1105.5165] [INSPIRE].

    ADS  Article  Google Scholar 

  2. J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231 [Int. J. Theor. Phys. 38 (1999) 1113] [hep-th/9711200] [INSPIRE].

    MathSciNet  ADS  MATH  Google Scholar 

  3. S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].

    MathSciNet  ADS  Article  Google Scholar 

  4. L. Randall and R. Sundrum, An alternative to compactification, Phys. Rev. Lett. 83 (1999) 4690 [hep-th/9906064] [INSPIRE].

    MathSciNet  ADS  MATH  Article  Google Scholar 

  5. S.S. Gubser, Breaking an abelian gauge symmetry near a black hole horizon, Phys. Rev. D 78 (2008) 065034 [arXiv:0801.2977] [INSPIRE].

    ADS  Google Scholar 

  6. S.A. Hartnoll, C.P. Herzog and G.T. Horowitz, Building a holographic superconductor, Phys. Rev. Lett. 101 (2008) 031601 [arXiv:0803.3295] [INSPIRE].

    ADS  Article  Google Scholar 

  7. H. Liu, J. McGreevy and D. Vegh, Non-Fermi liquids from holography, Phys. Rev. D 83 (2011) 065029 [arXiv:0903.2477] [INSPIRE].

    ADS  Google Scholar 

  8. M. Fujita, M. Kaminski and A. Karch, SL(2, \( \mathbb{Z} \)) Action on AdS/BCFT and Hall conductivities, JHEP 07 (2012) 150 [arXiv:1204.0012] [INSPIRE].

    MathSciNet  ADS  Article  Google Scholar 

  9. R. Laughlin, Quantized Hall conductivity in two-dimensions, Phys. Rev. B 23 (1981) 5632 [INSPIRE].

    ADS  Google Scholar 

  10. R. Laughlin, Anomalous quantum Hall effect: an incompressible quantum fluid with fractionallycharged excitations, Phys. Rev. Lett. 50 (1983) 1395 [INSPIRE].

    ADS  Article  Google Scholar 

  11. J. Jain, Composite fermion approach for the fractional quantum Hall effect, Phys. Rev. Lett. 63 (1989) 199 [INSPIRE].

    ADS  Article  Google Scholar 

  12. J. Jain, Theory of the fractional quantum Hall effect, Phys. Rev. B 41 (1990) 7653 [INSPIRE].

    ADS  Google Scholar 

  13. R.E. Prange and and S.M. Girvin, The quantum Hall effect, 2nd edition, Springer-Verlag, Berlin Germany (1990).

  14. X-G. Wen, Quantum field theory of many-body systems, Oxford University Press, Oxford U.K. (2004).

  15. R. Jackiw, Fractional charge and zero modes for planar systems in a magnetic field, Phys. Rev. D 29 (1984) 2375 [Erratum ibid. D 33 (1986) 2500] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  16. R. Jackiw, J. Avron, R. Seiler and B. Simon, Quantization of the Hall conductance for general multiparticle Schrödinger hamiltonians, Phys. Rev. Lett. 54 (1985) 259.

    MathSciNet  ADS  Article  Google Scholar 

  17. G.W. Semenoff and P. Sodano, Nonabelian adiabatic phases and the fractional quantum Hall effect, Phys. Rev. Lett. 57 (1986) 1195 [INSPIRE].

    MathSciNet  ADS  Article  Google Scholar 

  18. G.W. Semenoff, P. Sodano and Y.-S. Wu, Renormalization of the statistics parameter in three-dimensional electrodynamics, Phys. Rev. Lett. 62 (1989) 715 [INSPIRE].

    ADS  Article  Google Scholar 

  19. A.P. Polychronakos, Topological mass quantization and parity violation in (2 + 1)-dimensional QED, Nucl. Phys. B 281 (1987) 241 [INSPIRE].

    MathSciNet  ADS  Article  Google Scholar 

  20. A.P. Polychronakos, On the quantization of the coefficient of the abelian Chern-Simons term, Phys. Lett. B 241 (1990) 37 [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  21. N. Bralic, C. Fosco and F. Schaposnik, On the quantization of the Abelian Chern-Simons coefficient at finite temperature, Phys. Lett. B 383 (1996) 199 [hep-th/9509110] [INSPIRE].

    ADS  Google Scholar 

  22. C.L. Kane and M.P.A. Fisher, Quantized thermal transport in the fractional quantum Hall effect, Phys. Rev. B 55 (1997) 15832 [cond-mat/9603118].

    ADS  Google Scholar 

  23. B.I. Halperin, Quantized Hall conductance, current carrying edge states and the existence of extended states in a two-dimensional disordered potential, Phys. Rev. B 25 (1982) 2185 [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  24. S.A. Hartnoll and P. Kovtun, Hall conductivity from dyonic black holes, Phys. Rev. D 76 (2007) 066001 [arXiv:0704.1160] [INSPIRE].

    ADS  Google Scholar 

  25. S.A. Hartnoll, P.K. Kovtun, M. Muller and S. Sachdev, Theory of the Nernst effect near quantum phase transitions in condensed matter and in dyonic black holes, Phys. Rev. B 76 (2007) 144502 [arXiv:0706.3215] [INSPIRE].

    ADS  Google Scholar 

  26. J.L. Davis, P. Kraus and A. Shah, Gravity dual of a quantum Hall plateau transition, JHEP 11 (2008) 020 [arXiv:0809.1876] [INSPIRE].

    MathSciNet  ADS  Article  Google Scholar 

  27. 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] [INSPIRE].

    MathSciNet  ADS  Article  Google Scholar 

  28. O. Bergman, N. Jokela, G. Lifschytz and M. Lippert, Quantum Hall effect in a holographic model, JHEP 10 (2010) 063 [arXiv:1003.4965] [INSPIRE].

    ADS  Article  Google Scholar 

  29. O. Aharony, D. Marolf and M. Rangamani, Conformal field theories in Anti-de Sitter space, JHEP 02 (2011) 041 [arXiv:1011.6144] [INSPIRE].

    MathSciNet  ADS  Article  Google Scholar 

  30. M. Nozaki, T. Takayanagi and T. Ugajin, Central charges for BCFTs and holography, JHEP 06 (2012) 066 [arXiv:1205.1573] [INSPIRE].

    MathSciNet  ADS  Article  Google Scholar 

  31. M. Fujita, T. Takayanagi and E. Tonni, Aspects of AdS/BCFT, JHEP 11 (2011) 043 [arXiv:1108.5152] [INSPIRE].

    MathSciNet  ADS  Article  Google Scholar 

  32. M. Henningson and K. Skenderis, The holographic Weyl anomaly, JHEP 07 (1998) 023 [hep-th/9806087] [INSPIRE].

    MathSciNet  ADS  Article  Google Scholar 

  33. G. Hayward, Gravitational action for space-times with nonsmooth boundaries, Phys. Rev. D 47 (1993) 3275 [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  34. 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].

    MathSciNet  ADS  Google Scholar 

  35. E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].

    MathSciNet  ADS  MATH  Google Scholar 

  36. 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].

    MathSciNet  ADS  Article  Google Scholar 

  37. C. Herzog and D. Son, Schwinger-Keldysh propagators from AdS/CFT correspondence, JHEP 03 (2003) 046 [hep-th/0212072] [INSPIRE].

    MathSciNet  ADS  Article  Google Scholar 

  38. N.R. Cooper, B. I. Halperin and I.M. Ruzin, Thermoelectric response of an interacting two-dimensional electron gas in a quantizing magnetic field, Phys. Rev. B 55 (1997) 2344 [cond-mat/9607001].

    ADS  Google Scholar 

  39. T. Faulkner, H. Liu, J. McGreevy and D. Vegh, Emergent quantum criticality, Fermi surfaces and AdS 2, Phys. Rev. D 83 (2011) 125002 [arXiv:0907.2694] [INSPIRE].

    ADS  Google Scholar 

  40. O. Bergman, N. Jokela, G. Lifschytz and M. Lippert, Striped instability of a holographic Fermi-like liquid, JHEP 10 (2011) 034 [arXiv:1106.3883] [INSPIRE].

    ADS  Article  Google Scholar 

  41. D.B. Chklovskii, B.I. Shklovskii and L.I. Glazman, Electrostatics of edge channels, Phys. Rev. B 46 (1992) 4026.

    ADS  Google Scholar 

  42. N.B. Zhitenev, R.J. Haug, K. von Klitzing and K. Eberl, Time-resolved measurements of transport in edge channels, Phys. Rev. Lett. 71 (1993) 2292.

    ADS  Article  Google Scholar 

  43. S.W. Hwang, D.C. Tsui and M. Shayegan, Experimental evidence for finite-width edge channels in integer and fractional quantum Hall effects, Phys. Rev. B 48 (1993) 8161.

    ADS  Google Scholar 

  44. A.H. MacDonald, Edge states in the fractional-quantum-Hall-effect regime, Phys. Rev. Lett. 64 (1990) 220 [INSPIRE].

    ADS  Article  Google Scholar 

  45. X.G. Wen, Gapless boundary excitations in the quantum Hall states and in the chiral spin states, Phys. Rev. B 43 (1991) 11025 [INSPIRE].

    ADS  Google Scholar 

  46. X.G. Wen, Electrodynamical properties of gapless edge excitations in the fractional quantum Hall states, Phys. Rev. Lett. 64 (1990) 2206 [INSPIRE].

    ADS  Article  Google Scholar 

  47. X.G. Wen, Edge transport properties of the fractional quantum Hall states and weak-impurity scattering of a one-dimensional charge-density wave, Phys. Rev. B 44 (1991) 5708.

    ADS  Google Scholar 

  48. M.D. Johnson and A.H. MacDonald, Composite edges in the ν = 2/3 fractional quantum Hall effect, Phys. Rev. Lett. 67 (1991) 2060.

    ADS  Article  Google Scholar 

  49. V. Venkatachalam, S. Hart, L. Pfeiffer, K. West and A. Yacoby, Local thermometry of neutral modes on the quantum Hall edge, Nat. Phys. 8 (2012) 676 [arXiv:1202.6681].

    Article  Google Scholar 

  50. D. Melnikov, E. Orazi and P. Sodano, On the stability of the black hole solutions in AdS/BCFT models, to appear.

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Correspondence to Dmitry Melnikov.

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ArXiv ePrint: 1211.1416

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Melnikov, D., Orazi, E. & Sodano, P. On the AdS/BCFT approach to quantum Hall systems. J. High Energ. Phys. 2013, 116 (2013). https://doi.org/10.1007/JHEP05(2013)116

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  • DOI: https://doi.org/10.1007/JHEP05(2013)116

Keywords

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