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Spontaneous symmetry breaking of $\mathrm{SO}(2N)$ in Gross--Neveu theory from $2+ε$ expansion

Bilal Hawashin, Max Uetrecht

TL;DR

This work studies spontaneous symmetry breaking of $SO(2N)$ in the Gross–Neveu model by formulating a Fierz-complete renormalizable theory near the lower critical dimension $d=2{+}\epsilon$. By introducing $N_f$ Majorana flavors, it identifies four symmetry-allowed four-fermion couplings, analyzes their one-loop beta functions, and resolves three IR-stable fixed points: Gross–Neveu–Ising, symmetric-tensor, and adjoint-nematic. It shows that the symmetric-tensor fixed point becomes critical only above a flavor threshold $N_{f,c}^{ST}(N)\approx 0.56+1.48N$, while the adjoint-nematic fixed point coincides with the GNI point at $N_f=1$ and lacks a diverging order-parameter susceptibility at leading order. A two-sided Padé interpolation with upper-dimension results yields quantitative estimates for $N_{f,c}^{ST}$ in $d=3$, e.g., $N_{f,c}^{ST}(N{=}4)\approx 8.9$ and $N_{f,c}(N{=}8)\approx 16.1$, consistent with a second-order transition only in the GN–Ising channel. The findings corroborate a coherent picture across dimensional continuations and provide insight into symmetry-breaking transitions in Dirac-material contexts such as graphene and moiré systems.

Abstract

It was recently established that the paradigmatic Gross--Neveu model with $N$ copies of two-dimensional Dirac fermions features an $\mathrm{SO}(2N)$ 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 $\mathrm{SO}(2N)$ model predicts, besides the chiral Ising transition at $g_{c1}$, a second critical point $g_{c2}$ where $\mathrm{SO}(2N)$ is broken down to $\mathrm{SO}(N)\times\mathrm{SO}(N)$. A subsequent Wilsonian renormalization group analysis directly in $d=3$ supports its existence in a generalized theory, where $N_f$ copies of the $4N$-component Majorana fermions are introduced. This allows 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 $N_f = 1$, 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 $N_f = 1$, (ii) remains critical for all values of $N_f > N_{f,c}^{\mathrm{ST}}(N) \approx 0.56 + 1.48 N +\mathcal{O}(ε)$, which compares well with the estimate of previous studies. We further find that (iii) becomes equal to (i) when approaching $N_f = 1$. An instability is, however, only present in the susceptibility corresponding to the Gross--Neveu--Ising order parameter.

Spontaneous symmetry breaking of $\mathrm{SO}(2N)$ in Gross--Neveu theory from $2+ε$ expansion

TL;DR

This work studies spontaneous symmetry breaking of in the Gross–Neveu model by formulating a Fierz-complete renormalizable theory near the lower critical dimension . By introducing Majorana flavors, it identifies four symmetry-allowed four-fermion couplings, analyzes their one-loop beta functions, and resolves three IR-stable fixed points: Gross–Neveu–Ising, symmetric-tensor, and adjoint-nematic. It shows that the symmetric-tensor fixed point becomes critical only above a flavor threshold , while the adjoint-nematic fixed point coincides with the GNI point at and lacks a diverging order-parameter susceptibility at leading order. A two-sided Padé interpolation with upper-dimension results yields quantitative estimates for in , e.g., and , consistent with a second-order transition only in the GN–Ising channel. The findings corroborate a coherent picture across dimensional continuations and provide insight into symmetry-breaking transitions in Dirac-material contexts such as graphene and moiré systems.

Abstract

It was recently established that the paradigmatic Gross--Neveu model with copies of two-dimensional Dirac fermions features an symmetry if certain interactions are suppressed. This becomes evident when the theory is rewritten in terms of copies of two-dimensional Majorana fermions. Mean-field theory for the model predicts, besides the chiral Ising transition at , a second critical point where is broken down to . A subsequent Wilsonian renormalization group analysis directly in supports its existence in a generalized theory, where copies of the -component Majorana fermions are introduced. This allows 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 , 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 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 , (ii) remains critical for all values of , which compares well with the estimate of previous studies. We further find that (iii) becomes equal to (i) when approaching . An instability is, however, only present in the susceptibility corresponding to the Gross--Neveu--Ising order parameter.

Paper Structure

This paper contains 21 sections, 54 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Model
  3. Rotational invariance:
  4. $\mathrm{U}(1)$ charge symmetry:
  5. $\mathrm{SU}(N)$ symmetry:
  6. Parity symmetry:
  7. $\mathbb{Z}_2$ chiral symmetry:
  8. Renormalization group
  9. Symmetry-allowed operators
  10. Fierz identities for $N_f = 1$
  11. beta functions
  12. Fixed-point structure
  13. Gross--Neveu--Ising subspace:
  14. Adjoint-Nematic subspace:
  15. Critical exponents
  16. ...and 6 more sections

Figures (3)

  • Figure 1: Fixed-point coordinates of the $\mathrm{SO}(2N)$-symmetric-tensor fixed point as given in Eq. \ref{['eq:FPsymCoord']} for $N=4$ (left) and $N=8$ (right). The fixed point separates a $\mathrm{SO}(2N)$--symmetric phase and an ordered phase that breaks down ${\mathrm{SO}(2N) \to \mathrm{SO}(N) \times \mathrm{SO}(N)}$. While $N=8$ correspond to charge-neutral twisted bilayer graphene bistritzer2011moire, $N=4$ corresponds to a single sheet of graphene.
  • Figure 2: Anomalous dimension $\eta_\chi$ of the Majorana fermion for the $\mathrm{SO}(2N)$--symmetric theory in Eq. \ref{['eq:action']} at $N=4$ (left) and $N=8$ (right). The anomalous dimensions are evaluated at the Gross--Neveu--Ising (GNI) \ref{['eq:IsingFP']}, symmetric-tensor (ST) \ref{['eq:FPsymCoord']}, and adjoint-nematic fixed point (N) \ref{['eq:nematicFP']}.
  • Figure 3: Anomalous dimension $\eta_{\phi_n}$ of the Gross--Neveu--Ising, $\phi_1 = \langle \bar{\chi} (\mathbbm{1}_{2N} \otimes \mathbbm{1}_2) \chi \rangle$, symmetric tensor, $\phi_2 = \langle \bar{\chi} (\mathbb{S}_a \otimes \mathbbm{1}_2) \chi \rangle$, and adjoint nematic, $\phi_3 = \langle \bar{\chi} (\mathbb{A}_b \otimes \gamma_\mu \chi \rangle$, order parameters for the $\mathrm{SO}(2N)$--symmetric theory in Eq. \ref{['eq:action']}. We show the values of $x_n = \tfrac{1}{2} \left( 2 + \epsilon - \eta_{\phi_n} \right)$ for the $\mathrm{SO}(8)$-- (left) and $\mathrm{SO}(16)$--symmetric theory (right). At a critical point, the susceptibility of the corresponding order parameter $\phi_n$ diverges, requiring $x_n > \epsilon/2$, which is marked as a solid gray line. All anomalous dimensions are evaluated at the Gross--Neveu--Ising (GNI) \ref{['eq:IsingFP']}, symmetric-tensor (ST) \ref{['eq:FPsymCoord']}, and adjoint-nematic fixed point (N) \ref{['eq:nematicFP']}. While, at the symmetric-tensor fixed point, $x_1$ and $x_3$ never imply critical behavior, $x_2$ implies criticality only for $N_f > N_{f,c}^{\mathrm{ST}}(N=4) \approx 6.3$, or $N_f > N_{f,c}^{\mathrm{ST}}(N=8) \approx 12.5$ respectively. Probing the condition $x_2 > \epsilon/2$ for various integer values of $N$, we find $N_{f,c}^{\mathrm{ST}}(N) \approx 0.56 + 1.48 N +\mathcal{O}(\epsilon)$.