Journal of High Energy Physics

, 2015:147 | Cite as

Stress tensor and current correlators of interacting conformal field theories in 2+1 dimensions: fermionic Dirac matter coupled to U(1) gauge field

  • Yejin Huh
  • Philipp Strack
Open Access
Regular Article - Theoretical Physics


We compute the central charge C T and universal conductivity C J of N F fermions coupled to a U (1) gauge field up to next-to-leading order in the 1/N F expansion. We discuss implications of these precision computations as a diagnostic for response and entanglement properties of interacting conformal field theories for strongly correlated condensed matter phases and conformal quantum electrodynamics in 2 + 1 dimensions.


Conformal and W Symmetry 1/N Expansion 


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]
    K.G. Wilson and M.E. Fisher, Critical exponents in 3.99 dimensions, Phys. Rev. Lett. 28 (1972) 240 [INSPIRE].ADSCrossRefGoogle Scholar
  2. [2]
    R. Abe, Critical exponent η up to 1/N 2 for the three-dimensional system with short-range interaction, Prog. Theor. Phys. 49 (1973) 6.Google Scholar
  3. [3]
    A.C. Petkou, C T and C J up to next-to-leading order in 1/N in the conformally invariant O(N ) vector model for 2 < d < 4, Phys. Lett. B 359 (1995) 101 [hep-th/9506116] [INSPIRE].ADSCrossRefGoogle Scholar
  4. [4]
    S. El-Showk et al., Solving the 3D Ising model with the conformal bootstrap, Phys. Rev. D 86 (2012) 025022 [arXiv:1203.6064] [INSPIRE].ADSGoogle Scholar
  5. [5]
    S. El-Showk et al., Solving the 3D Ising model with the conformal bootstrap II. c-minimization and precise critical exponents, J. Stat. Phys. 157 (2014) 869 [arXiv:1403.4545] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  6. [6]
    M.C. Cha, et al., Universal conductivity of two-dimensional films at the superconductor-insulator transition, Phys. Rev. B 44 (1991) 6883.ADSCrossRefGoogle Scholar
  7. [7]
    R. Fazio and D. Zappala, ϵ expansion of the conductivity at the superconductor-Mott-insulator transitions, Phys. Rev. B 53 (1996) R8885 [cond-mat/9511004].ADSGoogle Scholar
  8. [8]
    S. Chakravarty, B.I. Halperin and D.R. Nelson, Two-dimensional quantum Heisenberg antiferromagnet at low temperatures, Phys. Rev. B 39 (1989) 2344 [INSPIRE].ADSCrossRefGoogle Scholar
  9. [9]
    A.V. Chubukov, S. Sachdev and J. Ye, Theory of two-dimensional quantum Heisenberg antiferromagnets with a nearly critical ground state, Phys. Rev. B 49 (1994) 11919 [INSPIRE].ADSCrossRefGoogle Scholar
  10. [10]
    R.K. Kaul and S. Sachdev, Quantum criticality of U(1) gauge theories with fermionic and bosonic matter in two spatial dimensions, Phys. Rev. B 77 (2008) 155105 [arXiv:0801.0723] [INSPIRE].ADSCrossRefGoogle Scholar
  11. [11]
    W. Chen, G.W. Semenoff and Y.-S. Wu, Two loop analysis of nonAbelian Chern-Simons theory, Phys. Rev. D 46 (1992) 5521 [hep-th/9209005] [INSPIRE].ADSMathSciNetGoogle Scholar
  12. [12]
    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].ADSCrossRefGoogle Scholar
  13. [13]
    T. Senthil et al., Deconfined quantum criticality, Science 303 (2004) 1490.ADSCrossRefGoogle Scholar
  14. [14]
    A.W. Sandvik, Evidence for deconfined quantum criticality in a two-dimensional Heisenberg model with four-spin interactions, Phys. Rev. Lett. 98 (2007) 227202 [cond-mat/0611343].ADSCrossRefGoogle Scholar
  15. [15]
    Y. Huh, P. Strack and S. Sachdev, Vector boson excitations near deconfined quantum critical points, Phys. Rev. Lett. 111 (2013) 166401 [arXiv:1307.6860] [INSPIRE].ADSCrossRefGoogle Scholar
  16. [16]
    W. Rantner and X.-G. Wen, Spin correlations in the algebraic spin liquid: implications for high-T c superconductors, Phys. Rev. B 66 (2002) 144501 [INSPIRE].ADSCrossRefGoogle Scholar
  17. [17]
    M. Franz, Z. Tesanovic and O. Vafek, QED 3 theory of pairing pseudogap in cuprates. 1. From D wave superconductor to antiferromagnet viaalgebraicFermi liquid, Phys. Rev. B 66 (2002) 054535 [cond-mat/0203333] [INSPIRE].ADSCrossRefGoogle Scholar
  18. [18]
    M. Franz, T. Pereg-Barnea, D.E. Sheehy and Z. Tesanovic, Gauge invariant response functions in algebraic Fermi liquids, Phys. Rev. B 68 (2003) 024508 [cond-mat/0211119] [INSPIRE].ADSCrossRefGoogle Scholar
  19. [19]
    R.K. Kaul, Y.B. Kaim, S. Sachdev and T. Senthil, Algebraic charge liquids, Nature Phys. 4 (2007) 28.ADSCrossRefGoogle Scholar
  20. [20]
    J. Cardy, Conformal field theory and statistical mechanics, arXiv:0807.3472.
  21. [21]
    A.M. Polyakov, Gauge fields and strings, Harwood Academic, Chur, Switzerland (1987).Google Scholar
  22. [22]
    S. Coleman, Aspects of symmetry, Cambridge University Press, Cambridge U.K. (1988).Google Scholar
  23. [23]
    T.W. Appelquist, M.J. Bowick, D. Karabali and L.C.R. Wijewardhana, Spontaneous chiral symmetry breaking in three-dimensional QED, Phys. Rev. D 33 (1986) 3704 [INSPIRE].ADSGoogle Scholar
  24. [24]
    T. Appelquist, D. Nash and L.C.R. Wijewardhana, Critical behavior in (2 + 1)-dimensional QED, Phys. Rev. Lett. 60 (1988) 2575 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  25. [25]
    D. Nash, Higher order corrections in (2 + 1)-dimensional QED, Phys. Rev. Lett. 62 (1989) 3024 [INSPIRE].ADSCrossRefGoogle Scholar
  26. [26]
    D.T. Son, Quantum critical point in graphene approached in the limit of infinitely strong Coulomb interaction, Phys. Rev. B 75 (2007) 235423 [cond-mat/0701501].ADSCrossRefGoogle Scholar
  27. [27]
    V. Juricic, O. Vafek and I.F. Herbut, Conductivity of interacting massless Dirac particles in graphene: collisionless regime, Phys. Rev. B 82 (2010) 235402 [arXiv:1009.3269] [INSPIRE].ADSCrossRefGoogle Scholar
  28. [28]
    I.F. Herbut and V. Mastropietro, Universal conductivity of graphene in the ultrarelativistic regime, Phys. Rev. B 87 (2013) 205445 [arXiv:1304.1988] [INSPIRE].ADSCrossRefGoogle Scholar
  29. [29]
    A.V. Kotikov and S. Teber, Two-loop fermion self-energy in reduced quantum electrodynamics and application to the ultra-relativistic limit of graphene, Phys. Rev. D 89 (2014) 065038 [arXiv:1312.2430] [INSPIRE].ADSGoogle Scholar
  30. [30]
    E. Barnes, E.H. Hwang, R. Throckmorton and S. Das Sarma, Effective field theory, three-loop perturbative expansion and their experimental implications in graphene many-body effects, Phys. Rev. B 89 (2014) 235431 [arXiv:1401.7011] [INSPIRE].ADSCrossRefGoogle Scholar
  31. [31]
    J. Braun, H. Gies, L. Janssen and D. Roscher, Phase structure of many-flavor QED 3, Phys. Rev. D 90 (2014) 036002 [arXiv:1404.1362] [INSPIRE].ADSGoogle Scholar
  32. [32]
    Y. Huh, P. Strack and S. Sachdev, Conserved current correlators of conformal field theories in 2 + 1 dimensions, Phys. Rev. B 88 (2013) 155109 [arXiv:1307.6863] [INSPIRE].ADSCrossRefGoogle Scholar
  33. [33]
    J. Cardy, The ubiquitousc: from the Stefan-Boltzmann law to quantum information, J. Stat. Mech. (2010) P10004.Google Scholar
  34. [34]
    E. Perlmutter, A universal feature of CFT Rényi entropy, JHEP 03 (2014) 117 [arXiv:1308.1083] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  35. [35]
    R.K. Kaul, R.G. Melko and A.W. Sandvik, Bridging lattice-scale physics and continuum field theory with quantum Monte Carlo simulations, Annu. Rev. Cond. Mat. Phys. 4 (2013) 179 [arXiv:1204.5405].ADSCrossRefGoogle Scholar
  36. [36]
    S.J. Hathrell, Trace anomalies and λϕ 4 theory in curved space, Annals Phys. 139 (1982) 136 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  37. [37]
    S.J. Hathrell, Trace anomalies and QED in curved space, Annals Phys. 142 (1982) 34 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  38. [38]
    I. Jack and H. Osborn, Background field calculations in curved spacetime: I. General application and application to scalar fields, Nucl. Phys. B 234 (1984) 331.ADSCrossRefGoogle Scholar
  39. [39]
    I. Jack, Background field calculations in curved space-time. 3. Application to a general gauge theory coupled to fermions and scalars, Nucl. Phys. B 253 (1985) 323 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  40. [40]
    A. Cappelli, D. Friedan and J.I. Latorre, C theorem and spectral representation, Nucl. Phys. B 352 (1991) 616 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  41. [41]
    H. Osborn and A.C. Petkou, Implications of conformal invariance in field theories for general dimensions, Annals Phys. 231 (1994) 311 [hep-th/9307010] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  42. [42]
    A. Petkou, Conserved currents, consistency relations and operator product expansions in the conformally invariant O(N ) vector model, Annals Phys. 249 (1996) 180 [hep-th/9410093] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  43. [43]
    M.F. Zoller and K.G. Chetyrkin, OPE of the energy-momentum tensor correlator in massless QCD, JHEP 12 (2012) 119 [arXiv:1209.1516] [INSPIRE].ADSCrossRefGoogle Scholar
  44. [44]
    D. Chowdhury, S. Raju, S. Sachdev, A. Singh and P. Strack, Multipoint correlators of conformal field theories: implications for quantum critical transport, Phys. Rev. B 87 (2013) 085138 [arXiv:1210.5247] [INSPIRE].ADSCrossRefGoogle Scholar
  45. [45]
    J.M. Maldacena and G.L. Pimentel, On graviton non-gaussianities during inflation, JHEP 09 (2011) 045 [arXiv:1104.2846] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  46. [46]
    I.R. Klebanov, S.S. Pufu, S. Sachdev and B.R. Safdi, Rényi entropies for free field theories, JHEP 04 (2012) 074 [arXiv:1111.6290] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  47. [47]
    A. Dymarsky, Z. Komargodski, A. Schwimmer and S. Theisen, On scale and conformal invariance in four dimensions, arXiv:1309.2921.
  48. [48]
    A. Bzowski and K. Skenderis, Comments on scale and conformal invariance in four dimensions, arXiv:1402.3208.
  49. [49]
    R.C. Myers and A. Sinha, Holographic c-theorems in arbitrary dimensions, JHEP 01 (2011) 125 [arXiv:1011.5819] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  50. [50]
    I.R. Klebanov, S.S. Pufu and B.R. Safdi, F-theorem without supersymmetry, JHEP 10 (2011) 038 [arXiv:1105.4598] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  51. [51]
    T. Appelquist, A.G. Cohen and M. Schmaltz, A new constraint on strongly coupled gauge theories, Phys. Rev. D 60 (1999) 045003 [hep-th/9901109] [INSPIRE].ADSGoogle Scholar
  52. [52]
    J.L. Cardy, Anisotropic corrections to correlation functions in finite size systems, Nucl. Phys. B 290 (1987) 355 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  53. [53]
    Tools and tables for quantum field theory calculations,
  54. [54]
    A. Bzowski, P. McFadden and K. Skenderis, Holographic predictions for cosmological 3-point functions, JHEP 03 (2012) 091 [arXiv:1112.1967] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  55. [55]
    A.I. Davydychev, A simple formula for reducing Feynman diagrams to scalar integrals, Phys. Lett. B 263 (1991) 107 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  56. [56]
    A.I. Davydychev, Recursive algorithm for evaluating vertex-type Feynman integrals, J. Phys. A 25 (1992) 5587.ADSMathSciNetMATHGoogle Scholar
  57. [57]
    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
  58. [58]
    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
  59. [59]
    O. Aharony et al., The thermal free energy in large N Chern-Simons-matter theories, JHEP 03 (2013) 121.ADSCrossRefGoogle Scholar

Copyright information

© The Author(s) 2015

Authors and Affiliations

  1. 1.Department of PhysicsHarvard UniversityCambridgeU.S.A.
  2. 2.Department of PhysicsUniversity of TorontoTorontoCanada
  3. 3.Institut für Theoretische PhysikUniversität zu KölnCologneGermany

Personalised recommendations