Beyond one-loop calculation: Higher-order effects on Gross-Neveu-Yukawa tensorial criticality

TL;DR

Using an $O(ε)$ expansion around $D=4$, this work analyzes the Gross-Neveu-Yukawa theory with an SO$(N)$ symmetric traceless tensor order parameter coupled to $N_f$ Majorana fermions in 2+1 dimensions. The authors derive two-loop RG flow equations for the dimensionless couplings $(α_g, λ_1, λ_2)$ and compute the $O(ε)$ corrections to the critical fermion numbers $N_{f,c1}$ and $N_{f,c2}$, finding that $N_{f,c1}$ increases while $N_{f,c2}$ decreases, thereby narrowing the window for a continuous transition. In the large-$N$ limit and for graphene-like $N=8$, they report a substantial reduction of $N_{f,c2}$ (≈34%) at two loops, and provide $O(ε^2)$ estimates of critical exponents and the mass-gap ratio, with three-loop corrections and resummations suggesting further reductions and highlighting the need for non-perturbative checks. The results yield concrete, testable predictions for Dirac materials and honeycomb systems and motivate cross-verification via numerical simulations or conformal bootstrap, while noting the asymptotic nature of the ε-series and the importance of resummation.

Abstract

We study the Gross-Neveu-Yukawa field theory for the SO($N$) symmetric traceless rank-two tensor order parameter coupled to Majorana fermions using the $ε$-expansion around upper critical dimensions of $3+1$ to two loops. Previously we established in the one-loop calculation that the theory does not exhibit a critical fixed point for $N \geq 4$, but that nevertheless the stable fixed point inevitably emerges at a large number of fermion flavors $N_f$. For $N_f < N_{f,c1} \approx N/2$, no critical fixed point exists; for $N_{f,c1} < N_f < N_{f,c2}$, a real critical fixed point emerges from the complex plane but fails to satisfy the additional stability conditions necessary for a continuous phase transition; and finally only for $N_f > N_{f,c2} \approx N$, the fixed point satisfies the stability conditions as well. In the present work we compute the $O(ε)$ (two-loop) corrections to the critical flavour numbers $N_{f,c1} $ and $N_{f,c2}$. Most importantly, we observe a sharp decrease in $N_{f,c2}$ from its one-loop value, which brings it closer to the point $N_f =1$ relevant to the standard Gross-Neveu model. Some three-loop results are also presented and discussed.

Figures (3)

• Figure 1: The plots for $N_{f,c1}/N$ and $N_{f,c2}/N$ as a function of $\epsilon$ in the $N\rightarrow\infty$ limit. (a) The plot for $N_{f,c1}/N$ and (b) the plot for $N_{f,c2}/N$. The red circles and blue diamonds stand for the $N_{f,c1}/N$ and $N_{f,c2}/N$ from the RG flow equations for given $\epsilon$ and $N\rightarrow\infty$ limit. The dashed lines are the fitting values from Eqs. \ref{['eq:nfc1']} and \ref{['eq:nfc2']}.
• Figure 2: The two-loop critical exponents and mass gap ratio in two-loop order with $\epsilon=1$. (a-b) The anomalous dimensions of fermions and order parameters, $\eta_{\psi}$ and $\eta_{\varphi}$, (c) the inverse correlation exponents $\nu^{-1}$, and the mass gap ratio.
• Figure 3: The comparison between one-loop and two-loop critical exponents and mass gap ratio in terms of $x=N_{f}-N_{f,c2}$, and fitted values from Eqs. \ref{['eq:quanti1']}-\ref{['eq:quanti4']} when $N=8$ with $\epsilon=1$. The red circle and blue triangle stand for the one-loop and two-loop results, respectively. And the red solid line and blue dashed line stand for the one-loop and two-loop fitted values from Eqs. \ref{['eq:quanti1']}-\ref{['eq:quanti4']}, respectively.