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Spontaneous symmetry breaking of SO(2N) in Gross-Neveu theory from 2+ε expansion

Bilal Hawashin1,* and Max Uetrecht2,†

  • *Contact author: hawashin@tp3.rub.de
  • †Contact author: max.uetrecht@tu-dortmund.de
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Phys. Rev. D 113, 065010 – Published 16 March, 2026

DOI: https://doi.org/10.1103/7yb8-7pk5

Abstract

It was recently established that the paradigmatic Gross-Neveu model with N copies of two-dimensional Dirac fermions features an SO(2N) symmetry if certain interactions are suppressed. This becomes evident when the theory is rewritten in terms of 2N copies of two-dimensional Majorana fermions. Mean-field theory for the SO(2N) model predicts, besides the chiral Ising transition at gc1, a second critical point gc2 where SO(2N) is broken down to SO(N)×SO(N). A subsequent Wilsonian renormalization group analysis directly in d=3 supports its existence in a generalized theory, where Nf copies of the 4N-component Majorana fermions are introduced. This allows us to track the evolution of a (i) quantum anomalous Hall Gross-Neveu-Ising, (ii) symmetric-tensor, and (iii) adjoint-nematic fixed point separately. However, it turns out that (ii) and (iii) lose their criticality when approaching Nf=1, suggesting that the transition is first order. In this work, we approach the problem from the lower-critical dimension of two. We construct a Fierz-complete renormalizable Lagrangian, compute the leading order β functions, fermion anomalous dimension, as well as the order parameter anomalous dimensions, and resolve the three universality classes corresponding to (i)–(iii). Before becoming equal to the Gaussian fixed point at Nf=1, (ii) remains critical for all values of Nf>Nf,cST(N)≈0.56+1.48N+O(ε), which compares well with the estimate of previous studies. We further find that (iii) becomes equal to (i) when approaching Nf=1. An instability is, however, only present in the susceptibility corresponding to the Gross-Neveu-Ising order parameter.

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Phys. Rev. D 113, 065010– Published 16 March, 2026
  • Received 31 October 2025
  • Accepted 25 February 2026
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