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

TL;DR

This paper investigates the spontaneous symmetry breaking of $ ext{SO}(2N)$ in the Gross--Neveu model using a $2+oldsymbol{ ext{epsilon}}$ expansion, revealing critical points, universality classes, and the nature of phase transitions.

Contribution

It constructs a Fierz-complete renormalizable Lagrangian and computes beta functions and anomalous dimensions to analyze symmetry breaking and universality classes in the model.

Findings

Identification of three universality classes (i, ii, iii) in the model.

Criticality of class (ii) persists for $N_f > N_{f,c}^{ ext{ST}}(N)$.

Transition becomes first order as $N_f$ approaches 1.

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 $\beta$ 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 $N_f > N_{f,c}^{\mathrm{ST}}(N) \approx 0.56 + 1.48 N +\mathcal{O}(\epsilon)$, 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.