  The semimetal to antiferromagnet quantum phase transition of the Hubbard
model on the honeycomb lattice has come to the forefront in the context of the
proposal that a semimetal to spin liquid transition can occur before the
transition to the antiferromagnetic phase. To study the semimetal to
antiferromagnet transition, we generalize the two-particle self-consistent
(TPSC) approach to the honeycomb lattice (a structure that can be realized in
graphene for example). We show that the critical interaction strength where the
transition occurs is $U_c/t=3.79\pm 0.01$ quite close to the value $U_c/t=3.869
\pm 0.013$ reported using large-scale quantum Monte Carlo simulations. This
reinforces the conclusion that the semimetal to spin liquid transition is
pre-empted by the transition to the antiferromagnet. Since TPSC satisfies the
Mermin-Wagner theorem, we find temperature-dependent results for the
antiferromagnetic and ferromagnetic correlation lengths as well as the
dependence of double occupancy and of the renormalized spin and charge
interactions on the bare interaction strength. We also estimate the value of
the crossover temperature to the renormalized classical regime as a function of
interaction strength.
