Bootstrapping 3D fermions with global symmetries

  • Luca Iliesiu
  • Filip Kos
  • David Poland
  • Silviu S. Pufu
  • David Simmons-Duffin
Open Access
Regular Article - Theoretical Physics


We study the conformal bootstrap for 4-point functions of fermions 〈ψ i ψ j ψ k ψ 〉 in parity-preserving 3d CFTs, where ψ i transforms as a vector under an O(N ) global symmetry. We compute bounds on scaling dimensions and central charges, finding features in our bounds that appear to coincide with the O(N ) symmetric Gross-Neveu-Yukawa fixed points. Our computations are in perfect agreement with the 1/N expansion at large N and allow us to make nontrivial predictions at small N . For values of N for which the Gross-Neveu-Yukawa universality classes are relevant to condensed-matter systems, we compare our results to previous analytic and numerical results.


1/N Expansion Conformal and W Symmetry Conformal Field Theory Global Symmetries 


Open Access

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  1. [1]
    A.M. Polyakov, Nonhamiltonian approach to conformal quantum field theory, Zh. Eksp. Teor. Fiz. 66 (1974) 23 [Sov. Phys. JETP 39 (1974) 9] [INSPIRE].
  2. [2]
    S. Ferrara, A.F. Grillo and R. Gatto, Tensor representations of conformal algebra and conformally covariant operator product expansion, Annals Phys. 76 (1973) 161 [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  3. [3]
    G. Mack, Duality in quantum field theory, Nucl. Phys. B 118 (1977) 445 [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  4. [4]
    S. El-Showk, M.F. Paulos, D. Poland, S. Rychkov, D. Simmons-Duffin and A. Vichi, Solving the 3D Ising model with the conformal bootstrap, Phys. Rev. D 86 (2012) 025022 [arXiv:1203.6064] [INSPIRE].MATHADSGoogle Scholar
  5. [5]
    S. El-Showk, M.F. Paulos, D. Poland, S. Rychkov, D. Simmons-Duffin and A. Vichi, 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].
  6. [6]
    F. Kos, D. Poland and D. Simmons-Duffin, Bootstrapping Mixed Correlators in the 3D Ising Model, JHEP 11 (2014) 109 [arXiv:1406.4858] [INSPIRE].CrossRefADSGoogle Scholar
  7. [7]
    D. Simmons-Duffin, A semidefinite program solver for the conformal bootstrap, JHEP 06 (2015) 174 [arXiv:1502.02033] [INSPIRE].CrossRefADSGoogle Scholar
  8. [8]
    F. Kos, D. Poland, D. Simmons-Duffin and A. Vichi, Precision islands in the Ising and O(N ) models, JHEP 08 (2016) 036 [arXiv:1603.04436] [INSPIRE].MathSciNetCrossRefMATHADSGoogle Scholar
  9. [9]
    D. Simmons-Duffin, The lightcone bootstrap and the spectrum of the 3D Ising CFT, JHEP 03 (2017) 086 [arXiv:1612.08471] [INSPIRE].MathSciNetCrossRefMATHADSGoogle Scholar
  10. [10]
    F. Kos, D. Poland and D. Simmons-Duffin, Bootstrapping the O(N ) vector models, JHEP 06 (2014) 091 [arXiv:1307.6856] [INSPIRE].CrossRefADSGoogle Scholar
  11. [11]
    F. Kos, D. Poland, D. Simmons-Duffin and A. Vichi, Bootstrapping the O(N ) archipelago, JHEP 11 (2015) 106 [arXiv:1504.07997] [INSPIRE].MathSciNetCrossRefMATHADSGoogle Scholar
  12. [12]
    D.J. Gross and A. Neveu, Dynamical symmetry breaking in asymptotically free field theories, Phys. Rev. D 10 (1974) 3235 [INSPIRE].ADSGoogle Scholar
  13. [13]
    L. Iliesiu, F. Kos, D. Poland, S.S. Pufu, D. Simmons-Duffin and R. Yacoby, Bootstrapping 3D fermions, JHEP 03 (2016) 120 [arXiv:1508.00012] [INSPIRE].CrossRefADSGoogle Scholar
  14. [14]
    J.A. Gracey, Anomalous mass dimension at O(1/N 2) in the O(N ) Gross-Neveu model, Phys. Lett. B 297 (1992) 293 [INSPIRE].CrossRefADSGoogle Scholar
  15. [15]
    S.E. Derkachov, N.A. Kivel, A.S. Stepanenko and A.N. Vasiliev, On calculation in 1/n expansions of critical exponents in the Gross-Neveu model with the conformal technique, hep-th/9302034 [INSPIRE].
  16. [16]
    J.A. Gracey, Computation of critical exponent η at O(1/N 3) in the four Fermi model in arbitrary dimensions, Int. J. Mod. Phys. A 9 (1994) 727 [hep-th/9306107] [INSPIRE].CrossRefADSGoogle Scholar
  17. [17]
    M. Moshe and J. Zinn-Justin, Quantum field theory in the large-N limit: a review, Phys. Rept. 385 (2003) 69 [hep-th/0306133] [INSPIRE].MathSciNetCrossRefMATHADSGoogle Scholar
  18. [18]
    A.C. Petkou, Operator product expansions and consistency relations in a O(N ) invariant fermionic CFT for 2 < d < 4, Phys. Lett. B 389 (1996) 18 [hep-th/9602054] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  19. [19]
    F. Parisen Toldin, M. Hohenadler, F.F. Assaad and I.F. Herbut, Fermionic quantum criticality in honeycomb and π-flux Hubbard models: finite-size scaling of renormalization-group-invariant observables from quantum Monte Carlo, Phys. Rev. B 91 (2015) 165108 [arXiv:1411.2502] [INSPIRE].CrossRefADSGoogle Scholar
  20. [20]
    I.F. Herbut, Interactions and phase transitions on graphene’s honeycomb lattice, Phys. Rev. Lett. 97 (2006) 146401 [cond-mat/0606195] [INSPIRE].
  21. [21]
    I.F. Herbut, V. Juricic and O. Vafek, Relativistic Mott criticality in graphene, Phys. Rev. B 80 (2009) 075432 [arXiv:0904.1019] [INSPIRE].CrossRefADSGoogle Scholar
  22. [22]
    M. Vojta, Y. Zhang and S. Sachdev, Quantum phase transitions in d-wave superconductors, Phys. Rev. Lett. 85 (2000) 4940 [INSPIRE].CrossRefADSGoogle Scholar
  23. [23]
    M. Vojta, Quantum phase transitions, Rep. Prog. Phys. 66 (2003) 2069 [cond-mat/0309604].
  24. [24]
    N. Bobev, S. El-Showk, D. Mazac and M.F. Paulos, Bootstrapping the three-dimensional supersymmetric Ising model, Phys. Rev. Lett. 115 (2015) 051601 [arXiv:1502.04124] [INSPIRE].CrossRefMATHADSGoogle Scholar
  25. [25]
    S.M. Chester, S. Giombi, L.V. Iliesiu, I.R. Klebanov, S.S. Pufu and R. Yacoby, Accidental symmetries and the conformal bootstrap, JHEP 01 (2016) 110 [arXiv:1507.04424] [INSPIRE].MathSciNetCrossRefMATHADSGoogle Scholar
  26. [26]
    S.M. Chester, L.V. Iliesiu, S.S. Pufu and R. Yacoby, Bootstrapping O(N ) vector models with four supercharges in 3 ≤ d ≤ 4, JHEP 05 (2016) 103 [arXiv:1511.07552] [INSPIRE].MathSciNetCrossRefMATHADSGoogle Scholar
  27. [27]
    T. Grover, D.N. Sheng and A. Vishwanath, Emergent space-time supersymmetry at the boundary of a topological phase, Science 344 (2014) 280 [arXiv:1301.7449] [INSPIRE].CrossRefADSGoogle Scholar
  28. [28]
    L. Fei, S. Giombi, I.R. Klebanov and G. Tarnopolsky, Yukawa CFTs and emergent supersymmetry, PTEP 2016 (2016) 12C105 [arXiv:1607.05316] [INSPIRE].
  29. [29]
    J.A. Gracey, Three loop calculations in the O(N ) Gross-Neveu model, Nucl. Phys. B 341 (1990) 403 [INSPIRE].CrossRefADSGoogle Scholar
  30. [30]
    B. Rosenstein, H.-L. Yu and A. Kovner, Critical exponents of new universality classes, Phys. Lett. B 314 (1993) 381 [INSPIRE].CrossRefADSGoogle Scholar
  31. [31]
    N. Zerf, C.-H. Lin and J. Maciejko, Superconducting quantum criticality of topological surface states at three loops, Phys. Rev. B 94 (2016) 205106 [arXiv:1605.09423] [INSPIRE].CrossRefADSGoogle Scholar
  32. [32]
    J.A. Gracey, T. Luthe and Y. Schröder, Four loop renormalization of the Gross-Neveu model, Phys. Rev. D 94 (2016) 125028 [arXiv:1609.05071] [INSPIRE].MathSciNetADSGoogle Scholar
  33. [33]
    L.N. Mihaila, N. Zerf, B. Ihrig, I.F. Herbut and M.M. Scherer, Gross-Neveu-Yukawa model at three loops and Ising critical behavior of Dirac systems, Phys. Rev. B 96 (2017) 165133 [arXiv:1703.08801] [INSPIRE].CrossRefADSGoogle Scholar
  34. [34]
    S. Giombi, V. Kirilin and E. Skvortsov, Notes on spinning operators in fermionic CFT, JHEP 05 (2017) 041 [arXiv:1701.06997] [INSPIRE].MathSciNetCrossRefMATHADSGoogle Scholar
  35. [35]
    F. Hofling, C. Nowak and C. Wetterich, Phase transition and critical behavior of the D = 3 Gross-Neveu model, Phys. Rev. B 66 (2002) 205111 [cond-mat/0203588] [INSPIRE].
  36. [36]
    B. Knorr, Ising and Gross-Neveu model in next-to-leading order, Phys. Rev. B 94 (2016) 245102 [arXiv:1609.03824] [INSPIRE].CrossRefADSGoogle Scholar
  37. [37]
    L. Janssen and I.F. Herbut, Antiferromagnetic critical point on graphene’s honeycomb lattice: a functional renormalization group approach, Phys. Rev. B 89 (2014) 205403 [arXiv:1402.6277] [INSPIRE].CrossRefADSGoogle Scholar
  38. [38]
    L. Kärkkäinen, R. Lacaze, P. Lacock and B. Petersson, Critical behavior of the three-dimensional Gross-Neveu and Higgs-Yukawa models, Nucl. Phys. B 415 (1994) 781 [Erratum ibid. B 438 (1995) 650] [hep-lat/9310020] [INSPIRE].
  39. [39]
    L. Wang, P. Corboz and M. Troyer, Fermionic quantum critical point of spinless fermions on a honeycomb lattice, New J. Phys. 16 (2014) 103008 [arXiv:1407.0029] [INSPIRE].CrossRefADSGoogle Scholar
  40. [40]
    Z.-X. Li, Y.-F. Jiang and H. Yao, Fermion-sign-free Majarana-quantum-Monte-Carlo studies of quantum critical phenomena of Dirac fermions in two dimensions, New J. Phys. 17 (2015) 085003 [arXiv:1411.7383] [INSPIRE].CrossRefADSGoogle Scholar
  41. [41]
    S. Chandrasekharan and A. Li, Quantum critical behavior in three dimensional lattice Gross-Neveu models, Phys. Rev. D 88 (2013) 021701 [arXiv:1304.7761] [INSPIRE].ADSGoogle Scholar
  42. [42]
    S. Hesselmann and S. Wessel, Thermal Ising transitions in the vicinity of two-dimensional quantum critical points, Phys. Rev. B 93 (2016) 155157 [arXiv:1602.02096] [INSPIRE].CrossRefADSGoogle Scholar
  43. [43]
    E.-G. Moon, C. Xu, Y.B. Kim and L. Balents, Non-Fermi liquid and topological states with strong spin-orbit coupling, Phys. Rev. Lett. 111 (2013) 206401 [arXiv:1212.1168] [INSPIRE].CrossRefADSGoogle Scholar
  44. [44]
    I.F. Herbut and L. Janssen, Topological Mott insulator in three-dimensional systems with quadratic band touching, Phys. Rev. Lett. 113 (2014) 106401 [arXiv:1404.5721] [INSPIRE].CrossRefADSGoogle Scholar
  45. [45]
    K. Diab, L. Fei, S. Giombi, I.R. Klebanov and G. Tarnopolsky, On C J and C T in the Gross-Neveu and O(N ) models, J. Phys. A 49 (2016) 405402 [arXiv:1601.07198] [INSPIRE].MATHGoogle Scholar
  46. [46]
    P. Kravchuk and D. Simmons-Duffin, Counting conformal correlators, arXiv:1612.08987 [INSPIRE].
  47. [47]
    D. Poland and D. Simmons-Duffin, Bounds on 4D conformal and superconformal field theories, JHEP 05 (2011) 017 [arXiv:1009.2087] [INSPIRE].MathSciNetCrossRefMATHADSGoogle Scholar
  48. [48]
    S. El-Showk and M.F. Paulos, Bootstrapping conformal field theories with the extremal functional method, Phys. Rev. Lett. 111 (2013) 241601 [arXiv:1211.2810] [INSPIRE].CrossRefADSGoogle Scholar
  49. [49]
    M. Heilmann, T. Hellwig, B. Knorr, M. Ansorg and A. Wipf, Convergence of derivative expansion in supersymmetric functional RG flows, JHEP 02 (2015) 109 [arXiv:1409.5650] [INSPIRE].CrossRefADSGoogle Scholar
  50. [50]
    L. Iliesiu, F. Kos, D. Poland, S.S. Pufu, D. Simmons-Duffin and R. Yacoby, Fermion-scalar conformal blocks, JHEP 04 (2016) 074 [arXiv:1511.01497] [INSPIRE].MathSciNetMATHADSGoogle Scholar
  51. [51]
    S. Sachdev, The landscape of the Hubbard model, in Proceedings, Theoretical Advanced Study Institute in Elementary Particle Physics (TASI 2010). String Theory and Its Applications: from MeV to the Planck scale, Boulder CO U.S.A., 1-25 June 2010, pg. 559 [arXiv:1012.0299] [INSPIRE].
  52. [52]
    K. Agashe, R. Contino and A. Pomarol, The minimal composite Higgs model, Nucl. Phys. B 719 (2005) 165 [hep-ph/0412089] [INSPIRE].
  53. [53]
    F. Caracciolo, A. Parolini and M. Serone, UV completions of composite Higgs models with partial compositeness, JHEP 02 (2013) 066 [arXiv:1211.7290] [INSPIRE].CrossRefADSGoogle Scholar
  54. [54]
    F. Caracciolo, A. Castedo Echeverri, B. von Harling and M. Serone, Bounds on OPE coefficients in 4D conformal field theories, JHEP 10 (2014) 020 [arXiv:1406.7845] [INSPIRE].MathSciNetCrossRefMATHADSGoogle Scholar

Copyright information

© The Author(s) 2018

Authors and Affiliations

  • Luca Iliesiu
    • 1
  • Filip Kos
    • 2
    • 3
  • David Poland
    • 4
  • Silviu S. Pufu
    • 1
  • David Simmons-Duffin
    • 5
    • 6
  1. 1.Joseph Henry LaboratoriesPrinceton UniversityPrincetonU.S.A.
  2. 2.Berkeley Center for Theoretical Physics, Department of PhysicsUniversity of CaliforniaBerkeleyU.S.A.
  3. 3.Theoretical Physics GroupLawrence Berkeley National LaboratoryBerkeleyU.S.A.
  4. 4.Department of PhysicsYale UniversityNew HavenU.S.A.
  5. 5.School of Natural SciencesInstitute for Advanced StudyPrincetonU.S.A.
  6. 6.Walter Burke Institute for Theoretical Physics, CaltechPasadenaU.S.A.

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