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

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 on-site 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 SO(3) order parameter ("chiral Heisenberg universality class"). Our calculation yields an estimate of the critical exponents $\nu \simeq 1.31$, $\eta_\phi \simeq 1.01$, and $\eta_\Psi \simeq 0.08$, 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 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 $\nu$ and $\eta_\phi$ and the double-digit percent range for $\eta_\Psi$. 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.