  We numerically investigate the quantum criticality of the chiral Heisenberg
universality class with the total number of fermion components $N$=8 in terms
of the Gross-Neveu theory. Auxiliary-field quantum Monte Carlo simulations are
performed for the square lattice Hubbard model in the presence of a $d$-wave
pairing field, inducing Dirac cones in the single particle spectrum. This
property makes the model particularly interesting because it turns out to
belong to the same universality class of the Hubbard model on the honeycomb
lattice, that is the canonical model for graphene, despite the unit cells being
apparently different (e.g. they contain one and two sites, respectively). We
indeed show that the two phase transitions, expected to occur on the square and
on the honeycomb lattices, have the same quantum criticality. We also argue
that details of the models, i.e., the way of counting $N$ and the anisotropy of
the Dirac cones, do not change the critical exponents. The present estimates of
the exponents for the $N$=8 chiral Heisenberg universality class are
$\nu$=1.05(5), $\eta_{\phi}$=0.75(4), and $\eta_{\psi}$=0.23(4), which are
compared with the previous numerical estimations.
