Antiferromagnetic critical point on graphene's honeycomb lattice: A functional renormalization group approach

Abstract

Electrons on the half-filled honeycomb lattice are expected to undergo a direct continuous transition from the semimetallic into the antiferromagnetic insulating phase with increase of onsite Hubbard repulsion. We attempt to further quantify the critical behavior at this quantum phase transition by means of functional renormalization group (RG), within an effective Gross-Neveu-Yukawa theory for an $\mathrm{SO}(3)$ order parameter (``chiral Heisenberg universality class''). Our calculation yields an estimate of the critical exponents $\ensuremath{\nu}\ensuremath{\simeq}1.31$, ${\ensuremath{\eta}}_{\ensuremath{\phi}}\ensuremath{\simeq}1.01$, and ${\ensuremath{\eta}}_{\ensuremath{\Psi}}\ensuremath{\simeq}0.08$, in reasonable agreement with the second-order expansion around the upper critical dimension. To test the validity of the present method, we use the conventional Gross-Neveu-Yukawa theory with ${\mathbb{Z}}_{2}$ order parameter (``chiral Ising universality class'') as a benchmark system. We explicitly show that our functional RG approximation in the sharp-cutoff scheme becomes one-loop exact both near the upper as well as the lower critical dimension. Directly in $2+1$ dimensions, our chiral Ising results agree with the best available predictions from other methods within the single-digit percent range for $\ensuremath{\nu}$ and ${\ensuremath{\eta}}_{\ensuremath{\phi}}$ and the double-digit percent range for ${\ensuremath{\eta}}_{\ensuremath{\Psi}}$. While one would expect a similar performance of our approximation in the chiral Heisenberg universality class, discrepancies with the results of other calculations here are more significant. Discussion and summary of various approaches is presented.