On Calculation of 1/n Expansions of Critical Exponents in the Gross-Neveu Model with the Conformal Technique

TL;DR

This work addresses the calculation of critical exponents in the Gross–Neveu model using conformal bootstrap at large $n$, and more broadly, establishes a general proof that a wide class of models at their RG fixed points exhibit critical conformal invariance for Green's functions. The authors construct a framework of Ward identities and breaking operators to determine when conformal symmetry is preserved, showing that dangerous operator contributions are absent or forbidden by symmetry in the GN setting. They provide explicit high-order results for the exponents $\eta$, $\Delta$, and $1/\nu$ in the $1/n$ expansion (notably $\eta$ to $O(1/n^3)$, and $\Delta$, $1/\nu$ to $O(1/n^2)$) using the conformal bootstrap, and discuss the applicability to related models via a two-coupling extension. The findings validate the conformal-invariance assumption underlying the bootstrap method and offer a practical pathway to compute higher-order critical dimensions in fermionic and scalar theories, with cross-checks against $2+\epsilon$ and $4-\epsilon$ expansions.

Abstract

A proof of critical conformal invariance of Green's functions for a quite wide class of models possessing critical scale invariance is given. A simple method for establishing critical conformal invariance of a composite operator, which has a certain critical dimension, is also presented. The method is illustrated with the example of the Gross--Neveu model and the exponents \et\ at order $1/n^3$, \Dl\ and $1/ν$ at order $1/n^2$ are calculated with the conformal bootstrap method.