Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 17 (2021), 064, 20 pages      arXiv:2102.12767
Contribution to the Special Issue on Algebraic Structures in Perturbative Quantum Field Theory in honor of Dirk Kreimer for his 60th birthday

Generalized Gross-Neveu Universality Class with Non-Abelian Symmetry

John A. Gracey
Theoretical Physics Division, Department of Mathematical Sciences, University of Liverpool, P.O. Box 147, Liverpool, L69 3BX, UK

Received February 26, 2021, in final form June 18, 2021; Published online June 29, 2021

We use the large $N$ critical point formalism to compute $d$-dimensional critical exponents at several orders in $1/N$ in an Ising Gross-Neveu universality class where the core interaction includes a Lie group generator. Specifying a particular symmetry group or taking the abelian limit of the final exponents recovers known results but also provides expressions for any Lie group or fermion representation.

Key words: critical exponents; large $N$ expansion; renormalization.

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  1. Ashkin J., Teller E., Statistics of two-dimensional lattices with four components, Phys. Rev. 64 (1943), 178-184.
  2. Assaad F.F., Herbut I.F., Pinning the order: the nature of quantum criticality in the Hubbard model on honeycomb lattice, Phys. Rev. X 3 (2013), 031010, 8 pages, arXiv:1304.6340.
  3. Bierenbaum I., Weinzierl S., The massless two-loop two-point function, Eur. Phys. J. C 32 (2003), 67-78, arXiv:hep-ph/0308311.
  4. Broadhurst D.J., Gracey J.A., Kreimer D., Beyond the triangle and uniqueness relations: non-zeta counterterms at large $N$ from positive knots, Z. Phys. C 75 (1997), 559-574, arXiv:hep-th/9607174.
  5. Broadhurst D.J., Kotikov A.V., Compact analytical form for non-zeta terms in critical exponents at order $1/N^3$, Phys. Lett. B 441 (1998), 345-353, arXiv:hep-th/9612013.
  6. Collins J.C., Vermaseren J.A.M., Axodraw Version 2, arXiv:1606.01177.
  7. D'eramo M., Peliti L., Parisi G., Theoretical predictions for critical exponents at the $\lambda$-point of bose liquids, Lett. Nuovo Cimento 2 (1971), 878-880.
  8. Derkachov S.É., Kivel N.A., Stepanenko A.S., Vasil'ev A.N., On calculation in $1/n$ expansions of critical exponents in the Gross-Neveu model with the conformal technique, arXiv:hep-th/9302034.
  9. Fei L., Giombi S., Klebanov I.R., Tarnopolsky G., Yukawa CFTs and emergent supersymmetry, Progr. Theoret. Exp. Phys. 2016 (2016), 12C105, 32 pages, arXiv:1607.05316.
  10. Gracey J.A., Three loop calculations in the $O(N)$ Gross-Neveu model, Nuclear Phys. B 341 (1990), 403-418.
  11. Gracey J.A., Calculation of exponent $\eta$ to $O\big(1/N^2\big)$ in the $O(N)$ Gross Neveu model, Internat. J. Modern Phys. A 6 (1991), 395-407, Erratum, Internat. J. Modern Phys. A 6 (1991), 2755.
  12. Gracey J.A., Computation of the three-loop $\beta$-function of the ${\rm O}(N)$ Gross-Neveu model in minimal subtraction, Nuclear Phys. B 367 (1991), 657-674.
  13. Gracey J.A., Anomalous mass dimension at $O\big(1/N^2\big)$ in the $O(N)$ Gross-Neveu model, Phys. Lett. B 297 (1992), 293-297.
  14. Gracey J.A., Computation of $\beta'(g_c)$ at $O\big(1/N^2\big)$ in the $O(N)$ Gross-Neveu model in arbitrary dimensions, Internat. J. Modern Phys. A 9 (1994), 567-589, arXiv:hep-th/9306106.
  15. Gracey J.A., Computation of critical exponent $\eta$ at $O\big(1/N^3\big)$ in the four-fermi model in arbitrary dimensions, Internat. J. Modern Phys. A 9 (1994), 727-744, arXiv:hep-th/9306107.
  16. Gracey J.A., Four loop $\overline{\rm MS}$ mass anomalous dimension in the Gross-Neveu model, Nuclear Phys. B 802 (2008), 330-350, arXiv:0804.1241.
  17. Gracey J.A., Large $N$ critical exponents for the chiral Heisenberg Gross-Neveu universality class, Phys. Rev. D 97 (2018), 105009, 17 pages, arXiv:1801.01320.
  18. Gracey J.A., Luthe T., Schröder Y., Four loop renormalization of the Gross-Neveu model, Phys. Rev. D 94 (2016), 125028, 18 pages, arXiv:1609.05071.
  19. Gross D.J., Neveu A., Dynamical symmetry breaking in asymptotically free field theories, Phys. Rev. D 10 (1974), 3235-3253.
  20. Herbut I.F., Juriucić V., Vafek O., Relativistic Mott criticality in graphene, Phys. Rev. B 80 (2009), 075432, 4 pages, arXiv:0904.1019.
  21. Ihrig B., Mihaila L.N., Scherer M.M., Critical behavior of Dirac fermions from perturbative renormalization, Phys. Rev. B 98 (2018), 125109, 20 pages, arXiv:1806.04977.
  22. Janssen L., Herbut I.F., Antiferromagnetic critical point on graphene's honeycomb lattice: A functional renormalization group approach, Phys. Rev. B 89 (2014), 205403, 14 pages, arXiv:1402.6277.
  23. Kärkkäinen L., Lacaze R., Lacock P., Petersson B., Critical behavior of the three-dimensional Gross-Neveu and Higgs-Yukawa models, Nuclear Phys. B 415 (1994), 781-796, arXiv:hep-lat/9310020.
  24. Kotikov A.V., The Gegenbauer polynomial technique: the evaluation of a class of Feynman diagrams, Phys. Lett. B 375 (1996), 240-248, arXiv:hep-ph/9512270.
  25. Ludwig A.W.W., Critical behavior of the two-dimensional random $q$-state Potts model by expansion in $(q-2)$, Nuclear Phys. B 285 (1987), 97-142.
  26. Luperini C., Rossi P., Three-loop $\beta$ function(s) and effective potential in the Gross-Neveu model, Ann. Physics 212 (1991), 371-401.
  27. Mihaila L.N., Zerf N., Ihrig B., Herbut I.F., Scherer M.M., Gross-Neveu-Yukawa model at three loops and Ising critical behavior of Dirac systems, Phys. Rev. B 96 (2017), 165133, 6 pages, arXiv:1703.08801.
  28. Parisi G., On self-consistency conditions in conformal covariant field theory, Lett. Nuovo Cimento 4 (1972), 777-780.
  29. Parisi G., Peliti L., Calculation of critical indices, Lett. Nuovo Cimento 2 (1971), 627-628.
  30. Polyakov A.M., Microscopic description of critical phenomena, Sov. Phys. JETP 28 (1969), 533-539.
  31. Polyakov A.M., Conformal symmetry of critical fluctuations, JETP Lett. 12 (1970), 381-383.
  32. Ray S., Ihrig B., Kruti D., Gracey J.A., Scherer M.M., Janssen L., Fractionalized quantum criticality in spin-orbital liquids from field theory beyond the leading order, Phys. Rev. B 103 (2021), 155160, 18 pages, arXiv:2101.10335.
  33. Rosenstein B., Warr B.J., Park S.H., Four-fermion theory is renormalizable in 2+1 dimensions, Phys. Rev. Lett. 62 (1989), 1433-1436.
  34. Rosenstein B., Warr B.J., Park S.H., Dynamical symmetry breaking in four-fermion interaction models, Phys. Rep. 205 (1991), 59-108.
  35. Seifert U.F.P., Dong X.Y., Chulliparambil S., Vojta M., Tu H.H., Janssen L., Fractionalized fermionic quantum criticality in spin-orbital Mott insulators, Phys. Rev. Lett. 125 (2020), 257202, 7 pages, arXiv:2009.05051.
  36. Sorella S., Otsuka Y., Yunoki S., Absence of a spin liquid phase in the Hubbard model on the honeycomb lattice, Sci. Rep. 2 (2012), 992, 5 pages, arXiv:1207.1783.
  37. Tentyukov M., Vermaseren J.A.M., The multithreaded version of FORM, Comput. Phys. Comm. 181 (2010), 1419-1427, arXiv:hep-ph/0702279.
  38. van Ritbergen T., Schellekens A.N., Vermaseren J.A.M., Group theory factors for Feynman diagrams, Internat. J. Modern Phys. A 14 (1999), 41-96, arXiv:hep-ph/9802376.
  39. Vasil'ev A.N., Derkachov S.É., Kivel N.A., Stepanenko A.S., The $1/n$ expansion in the Gross-Neveu model: conformal bootstrap calculation of the index $\eta$ in order $1/n^3$, Theoret. and Math. Phys. 94 (1993), 127-136.
  40. Vasil'ev A.N., Nalimov M.Yu., Analog of dimensional regularization for calculation of the renormalization group functions in the $1/n$ expansion for arbitrary dimension of space, Theoret. and Math. Phys. 55 (1983), 423-431.
  41. Vasil'ev A.N., Nalimov M.Yu., The $CP^{N-1}$ model: calculation of anomalous dimensions and the mixing matrices in the order $1/N$, Theoret. and Math. Phys. 56 (1983), 643-653.
  42. Vasil'ev A.N., Pismak Yu.M., Honkonen J.R., $1/n$ expansion: calculation of the exponents $\eta$ and $\nu$ in the order $1/n^2$ for arbitrary number of dimensions, Theoret. and Math. Phys. 47 (1981), 465-475.
  43. Vasil'ev A.N., Pismak Yu.M., Honkonen J.R., Simple method of calculating the critical indices in the $1/n$ expansion, Theoret. and Math. Phys. 46 (1981), 104-113.
  44. Vasil'ev A.N., Pismak Yu.M., Honkonen J.R., $1/n$ expansion: calculation of the exponent $\eta$ in the order $1/n^3$ by the conformal bootstrap method, Theoret. and Math. Phys. 50 (1982), 127-134.
  45. Vasil'ev A.N., Stepanenko A.S., $1/n$-expansion in the Gross-Neveu model: calculation of the $1/\nu$ index to the order $1/n^2$ by the conformal bootstrap method, Theoret. and Math. Phys. 97 (1993), 1349-1354.
  46. Vermaseren J.A.M., New features of FORM, arXiv:math-ph/0010025.
  47. Wetzel W., Two-loop $\beta$-function for the Gross-Neveu model, Phys. Lett. B 153 (1985), 297-299.
  48. Wilson K.G., Renormalization group and critical phenomena. II. Phase-space cell analysis of critical behavior, Phys. Rev. B 4 (1971), 3184-3205.
  49. Wilson K.G., Feynman graph expansion for critical exponents, Phys. Rev. Lett. 28 (1972), 548-551.
  50. Wilson K.G., Fisher M.E., Critical exponents in 3.99 dimensions, Phys. Rev. Lett. 28 (1972), 240-243.
  51. Wilson K.G., Kogut J., The renormalization group and the $\epsilon$ expansion, Phys. Rep. 12 (1974), 75-199.
  52. Zerf N., Mihaila L.N., Marquard P., Herbut I.F., Scherer M.M., Four-loop critical exponents for the Gross-Neveu-Yukawa models, Phys. Rev. D 96 (2017), 096010, 19 pages, arXiv:1709.05057.
  53. Zinn-Justin J., Four-fermion interaction near four dimensions, Nuclear Phys. B 367 (1991), 105-122.

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