Gross-Neveu-Heisenberg criticality from $2+\boldsymbol{\epsilon}$ expansion

TL;DR

This paper develops an epsilon expansion approach around two dimensions to analyze the Gross-Neveu-Heisenberg quantum critical point, providing improved estimates for critical exponents relevant to graphene and related materials.

Contribution

It introduces a Fierz-complete epsilon expansion around two dimensions to study the Gross-Neveu-Heisenberg universality class, complementing previous methods and accounting for multiple interaction channels.

Findings

Improved estimates for critical exponents in 2+1 dimensions.

Identification of a small correction-to-scaling exponent for graphene.

Highlighting the importance of multiple interaction channels at criticality.

Abstract

The Gross-Neveu-Heisenberg universality class describes a continuous quantum phase transition between a Dirac semimetal and an antiferromagnetic insulator. Such quantum critical points have originally been discussed in the context of Hubbard models on $\pi$-flux and honeycomb lattices, but more recently also in Bernal-stacked bilayer models, of potential relevance for bilayer graphene. Here, we demonstrate how the critical behavior of this fermionic universality class can be computed within an $\epsilon$ expansion around the lower critical space-time dimension of two. This approach is complementary to the previously studied expansion around the upper critical dimension of four. The crucial technical novelty near the lower critical dimension is the presence of different four-fermion interaction channels at the critical point, which we take into account in a Fierz-complete way. By interpolating between the lower and upper critical dimensions, we obtain improved estimates for the critical exponents in 2+1 space-time dimensions. For the situation relevant to single-layer graphene, we find an unusually small leading-correction-to-scaling exponent, arising from the competition between different interaction channels. This suggests that corrections to scaling may need to be taken into account when comparing analytical estimates with numerical data from finite-size extrapolations.