  We investigate a quantum criticality of an antiferromagnetic phase transition
in the Hubbard model on a square lattice with a $d$-wave pairing field by
large-scale auxiliary-field quantum Monte Carlo simulations. Since the $d$-wave
pairing filed induces Dirac cones in the non-interacting single-particle
spectrum, the quantum criticality should correspond to the chiral Heisenberg
universality class in terms of the Gross-Neveu theory, which is the same as
those expected in the Hubbard model on the honeycomb lattice, despite the unit
cells being different (e.g., they contain one and two sites, respectively). We
show that both the two phase transitions, expected to occur on the square and
on the honeycomb lattices, indeed have the same quantum criticality. We also
argue that details of the models, i.e., the way of counting the total number
$N$ of fermion components and the anisotropy of the Dirac cones, do not change
the critical exponents.
