Four-loop critical exponents for the Gross-Neveu-Yukawa models

TL;DR

This paper advances the perturbative RG analysis of Gross-Neveu-Yukawa models by deriving full four-loop beta and gamma functions in $4-\epsilon$ dimensions for the chiral Ising, XY, and Heisenberg classes and extracting critical exponents to $\mathcal{O}(\epsilon^4)$. It provides Pad\é estimates to 2+1 dimensions, discusses emergent supersymmetry at specific $N$ (notably $N=1/4$ for Ising and $N=1/2$ for XY), and connects results to Dirac/Weyl semimetal transitions in graphene and topological-insulator surfaces. The study includes extensive computational details, cross-checks against known limits, and critical comparisons with FRG, bootstrap, and Monte Carlo data, highlighting both agreements and tensions that motivate future resummation and methodology refinements. Overall, the work significantly strengthens the quantitative understanding of 2+1D quantum criticality in Dirac systems and the role of higher-loop corrections in determining universal exponents. $1/\nu$, $\eta_\phi$, and $\omega$ are computed with high precision, and SUSY-consistent relations are demonstrated in several cases, underscoring the intricate link between symmetry, dimensionality, and critical behavior in GNY theories.

Abstract

We study the chiral Ising, the chiral XY and the chiral Heisenberg models at four-loop order with the perturbative renormalization group in $4-ε$ dimensions and compute critical exponents for the Gross-Neveu-Yukawa fixed points to order $\mathcal{O}(ε^4)$. Further, we provide Padé estimates for the correlation length exponent, the boson and fermion anomalous dimension as well as the leading correction to scaling exponent in 2+1 dimensions. We also confirm the emergence of supersymmetric field theories at four loops for the chiral Ising and the chiral XY models with $N=1/4$ and $N=1/2$ fermions, respectively. Furthermore, applications of our results relevant to various quantum transitions in the context of Dirac and Weyl semimetals are discussed, including interaction-induced transitions in graphene and surface states of topological insulators.

Figures (1)

• Figure 1: Sample diagrams contributing to the fermion-fermion-scalar renormalization in $D=3$ dimensions. The scalar field is denoted $\phi$ and the Majorana fermions $\Psi$.