Transport in Chern-Simons-matter theories

  • Guy Gur-Ari
  • Sean Hartnoll
  • Raghu Mahajan
Open Access
Regular Article - Theoretical Physics


The frequency-dependent longitudinal and Hall conductivities — σ xx and σ xy — are dimensionless functions of ω/T in 2+1 dimensional CFTs at nonzero temperature. These functions characterize the spectrum of charged excitations of the theory and are basic experimental observables. We compute these conductivities for large N Chern-Simons theory with fermion matter. The computation is exact in the ’t Hooft coupling λ at N = ∞. We describe various physical features of the conductivity, including an explicit relation between the weight of the delta function at ω = 0 in σ xx and the existence of infinitely many higher spin conserved currents in the theory. We also compute the conductivities perturbatively in Chern-Simons theory with scalar matter and show that the resulting functions of ω/T agree with the strong coupling fermionic result. This provides a new test of the conjectured 3d bosonization duality. In matching the Hall conductivities we resolve an outstanding puzzle by carefully treating an extra anomaly that arises in the regularization scheme used.


1/N Expansion Chern-Simons Theories Duality in Gauge Field Theories Field Theories in Lower Dimensions 


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. Luttinger, Theory of thermal transport coefficients, Phys. Rev. 135 (1964) A1505 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  2. [2]
    D. Forster, Hydrodynamics, fluctuations, broken symmetry, and correlation functions, Perseus books, U.S.A. (1975).Google Scholar
  3. [3]
    S. Sachdev, Quantum phase transitions, 2nd ed., Cambridge University Press, Cambridge U.K. (2011).Google Scholar
  4. [4]
    K. Damle and S. Sachdev, Non-zero temperature transport near quantum critical points, Phys. Rev. B 56 (1997) 8714 [cond-mat/9705206] [INSPIRE].
  5. [5]
    S. Sachdev, Nonzero temperature transport near fractional quantum Hall critical points, Phys. Rev. B 57 (1998) 7157 [cond-mat/9709243].
  6. [6]
    C.P. Herzog, P. Kovtun, S. Sachdev and D.T. Son, Quantum critical transport, duality and M-theory, Phys. Rev. D 75 (2007) 085020 [hep-th/0701036] [INSPIRE].ADSMathSciNetGoogle Scholar
  7. [7]
    J. Maldacena and A. Zhiboedov, Constraining conformal field theories with a higher spin symmetry, J. Phys. A 46 (2013) 214011 [arXiv:1112.1016] [INSPIRE].ADSMathSciNetMATHGoogle Scholar
  8. [8]
    J. Maldacena and A. Zhiboedov, Constraining conformal field theories with a slightly broken higher spin symmetry, Class. Quant. Grav. 30 (2013) 104003 [arXiv:1204.3882] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  9. [9]
    O. Aharony, G. Gur-Ari and R. Yacoby, Correlation functions of large-N Chern-Simons-matter theories and bosonization in three dimensions, JHEP 12 (2012) 028 [arXiv:1207.4593] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  10. [10]
    O. Aharony, S. Giombi, G. Gur-Ari, J. Maldacena and R. Yacoby, The thermal free energy in large-N Chern-Simons-matter theories, JHEP 03 (2013) 121 [arXiv:1211.4843] [INSPIRE].ADSCrossRefGoogle Scholar
  11. [11]
    G. Gur-Ari and R. Yacoby, Correlators of large-N fermionic Chern-Simons vector models, JHEP 02 (2013) 150 [arXiv:1211.1866] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  12. [12]
    X.G. Wen, Non-Abelian statistics in the fractional quantum Hall states, Phys. Rev. Lett. 66 (1991) 802 [INSPIRE].
  13. [13]
    E.H. Fradkin, C. Nayak, A. Tsvelik and F. Wilczek, A Chern-Simons effective field theory for the Pfaffian quantum Hall state, Nucl. Phys. B 516 (1998) 704 [cond-mat/9711087] [INSPIRE].
  14. [14]
    X.-G. Wen, Projective construction of non-Abelian quantum Hall liquids, Phys. Rev. B 60 (1999) 8827 [cond-mat/9811111] [INSPIRE].
  15. [15]
    N. Seiberg and E. Witten, Gapped boundary phases of topological insulators via weak coupling, arXiv:1602.04251 [INSPIRE].
  16. [16]
    T.H. Hansson, M. Hermanns, S.H. Simon and S.F. Viefers, Quantum Hall hierarchies, arXiv:1601.01697 [INSPIRE].
  17. [17]
    N. Dorey, D. Tong and C. Turner, A matrix model for non-Abelian quantum Hall states, arXiv:1603.09688 [INSPIRE].
  18. [18]
    W. Chen, M.P.A. Fisher and Y.-S. Wu, Mott transition in an anyon gas, Phys. Rev. B 48 (1993) 13749 [cond-mat/9301037] [INSPIRE].
  19. [19]
    S.L. Sondhi, S.M. Girvin, J.P. Carini and D. Shahar, Continuous quantum phase transitions, Rev. Mod. Phys. 69 (1997) 315 [INSPIRE].ADSCrossRefGoogle Scholar
  20. [20]
    M. Barkeshli and X.G. Wen, Effective field theory and projective construction for Z k parafermion fractional quantum Hall states, Phys. Rev. B 81 (2010) 155302 [arXiv:0910.2483].ADSCrossRefGoogle Scholar
  21. [21]
    A. Vaezi and M. Barkeshli, Fibonacci anyons from Abelian bilayer quantum Hall states, Phys. Rev. Lett. 113 (2014) 236804 [arXiv:1403.3383] [INSPIRE].ADSCrossRefGoogle Scholar
  22. [22]
    D.J. Clarke and C. Nayak, Chern-Simons-Higgs transitions out of topological superconducting phases, Phys. Rev. B 92 (2015) 155110 [Erratum ibid. B 93 (2016) 119907] [arXiv:1507.00344] [INSPIRE].
  23. [23]
    M. Geracie, M. Goykhman and D.T. Son, Dense Chern-Simons matter with fermions at large-N , JHEP 04 (2016) 103 [arXiv:1511.04772] [INSPIRE].ADSCrossRefGoogle Scholar
  24. [24]
    S. Giombi, S. Minwalla, S. Prakash, S.P. Trivedi, S.R. Wadia and X. Yin, Chern-Simons theory with vector fermion matter, Eur. Phys. J. C 72 (2012) 2112 [arXiv:1110.4386] [INSPIRE].ADSCrossRefGoogle Scholar
  25. [25]
    O. Aharony, G. Gur-Ari and R. Yacoby, D = 3 bosonic vector models coupled to Chern-Simons gauge theories, JHEP 03 (2012) 037 [arXiv:1110.4382] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  26. [26]
    A. Zee, Quantum Hall fluids, in Field theory, topology and condensed matter physics, Springer, Germany (1995), pg 99 [cond-mat/9501022] [INSPIRE].
  27. [27]
    L. Di Pietro and Z. Komargodski, private communication.Google Scholar
  28. [28]
    D. Radičević, Disorder operators in Chern-Simons-fermion theories, JHEP 03 (2016) 131 [arXiv:1511.01902] [INSPIRE].ADSGoogle Scholar
  29. [29]
    O. Aharony, Baryons, monopoles and dualities in Chern-Simons-matter theories, JHEP 02 (2016) 093 [arXiv:1512.00161] [INSPIRE].
  30. [30]
    S.A. Hartnoll, Lectures on holographic methods for condensed matter physics, Class. Quant. Grav. 26 (2009) 224002 [arXiv:0903.3246] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  31. [31]
    F. Ferrari, The analytic renormalization group, Nucl. Phys. B 909 (2016) 880 [arXiv:1602.07355] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  32. [32]
    D.R. Gulotta, C.P. Herzog and M. Kaminski, Sum rules from an extra dimension, JHEP 01 (2011) 148 [arXiv:1010.4806] [INSPIRE].
  33. [33]
    W. Witczak-Krempa and S. Sachdev, The quasi-normal modes of quantum criticality, Phys. Rev. B 86 (2012) 235115 [arXiv:1210.4166] [INSPIRE].ADSCrossRefGoogle Scholar
  34. [34]
    E. Katz, S. Sachdev, E.S. Sørensen and W. Witczak-Krempa, Conformal field theories at nonzero temperature: operator product expansions, Monte Carlo and holography, Phys. Rev. B 90 (2014) 245109 [arXiv:1409.3841] [INSPIRE].ADSCrossRefGoogle Scholar
  35. [35]
    P. Mazur, Non-ergodicity of phase functions in certain systems, Physica 43 (1969) 533.ADSMathSciNetCrossRefGoogle Scholar
  36. [36]
    M. Suzuki, Ergodicity, constants of motion, and bounds for susceptibilties, Physica 51 (1971) 277.ADSMathSciNetCrossRefGoogle Scholar
  37. [37]
    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].
  38. [38]
    S.A. Hartnoll and D.M. Hofman, Locally critical resistivities from Umklapp scattering, Phys. Rev. Lett. 108 (2012) 241601 [arXiv:1201.3917] [INSPIRE].ADSCrossRefGoogle Scholar
  39. [39]
    C. Closset, T.T. Dumitrescu, G. Festuccia, Z. Komargodski and N. Seiberg, Comments on Chern-Simons contact terms in three dimensions, JHEP 09 (2012) 091 [arXiv:1206.5218] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  40. [40]
    A.N. Redlich, Parity violation and gauge noninvariance of the effective gauge field action in three-dimensions, Phys. Rev. D 29 (1984) 2366 [INSPIRE].ADSMathSciNetGoogle Scholar
  41. [41]
    V.P. Spiridonov and F.V. Tkachov, Two loop contribution of massive and massless fields to the Abelian Chern-Simons term, Phys. Lett. B 260 (1991) 109 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  42. [42]
    R.A. Davison, L.V. Delacrétaz, B. Goutéraux and S.A. Hartnoll, Hydrodynamic theory of quantum fluctuating superconductivity, arXiv:1602.08171 [INSPIRE].
  43. [43]
    P. Kovtun and L.G. Yaffe, Hydrodynamic fluctuations, long time tails and supersymmetry, Phys. Rev. D 68 (2003) 025007 [hep-th/0303010] [INSPIRE].ADSGoogle Scholar

Copyright information

© The Author(s) 2016

Authors and Affiliations

  1. 1.Stanford Institute for Theoretical PhysicsStanford UniversityStanfordU.S.A.

Personalised recommendations