Dirac electrons in the square lattice Hubbard model with a $d$-wave pairing field: chiral Heisenberg universality class revisited

TL;DR

This study uses quantum Monte Carlo simulations to analyze the quantum criticality of a Dirac fermion system with a $d$-wave pairing field, revealing universal behavior across different lattice models relevant to graphene.

Contribution

It demonstrates that the square lattice Hubbard model with a $d$-wave pairing field belongs to the same universality class as the honeycomb lattice Hubbard model, providing new insights into quantum critical behavior.

Findings

Critical exponents estimated: $ u$=1.05(5), $ ext{eta}_ ext{phi}$=0.75(4), $ ext{eta}_ ext{psi}$=0.23(4)

Square and honeycomb lattice models share the same quantum criticality

Model details like $N$ counting and Dirac cone anisotropy do not affect critical exponents

Abstract

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.