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QMC study of the chiral Heisenberg Gross-Neveu universality class

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.