Nature of the antiferromagnetic quantum phase transition on the honeycomb lattice

We address the nature of the antiferromagnetic quantum phase transition that separates a semimetal from an antiferromagnet in the repulsive Hubbard model defined on the honeycomb lattice. At the critical point, the fermions acquire an anomalous dimension $\eta$ due to their strong coupling to the fluctuations of the order parameter $\phi$. The finite $\eta$ in turn induces a singular $\phi^{4}$ term and a non-analytical spin susceptibility signaling the breakdown of Hertz's $\phi^{4}$ theory. As a result, the continuous antiferromagnetic quantum phase transition is internally unstable and turns into a first order transition.