Chiral Heisenberg Gross-Neveu-Yukawa criticality: Honeycomb vs. SLAC fermions

TL;DR

The work addresses the critical properties of the chiral Heisenberg Gross-Neveu-Yukawa transition for N=8 Dirac components in 2+1D, using large-scale quantum Monte Carlo with SLAC fermions and comparing to honeycomb lattices. By implementing an N-pole SLAC Dirac cone and applying projector auxiliary-field QMC, the authors show reduced finite-size effects and extract critical exponents $\nu=1.02(3)$, $\eta_\phi=0.73(1)$, and $\eta_\psi=0.09(1)$ that agree between lattice types at large $L$, consistent with the GNY universality class. They analyze weak- and strong-coupling limits, showing a Dirac semimetal to AFM transition at $U_c\approx 6$ with gapless Goldstone modes in the AFM phase and isotropic critical spin dynamics near $U_c$, while confirming non-perturbative renormalizability through finite-size scaling. The study demonstrates that SLAC fermions, despite their nonlocal hopping, yield reliable critical behavior with improved finite-size scaling and may be extended to other Dirac systems and interactions. Overall, the results reinforce the universality of the chiral Heisenberg GNY fixed point and highlight the practical advantages of the SLAC formulation for QMC studies of Dirac-criticality.

Abstract

We perform large scale quantum Monte Carlo simulations of the Hubbard model at half filling with a single Dirac cone close to the critical point, which separates a Dirac semi-metal from an antiferromagnetically ordered phase where SU(2) spin rotational symmetry is spontaneously broken. We discuss the implementation of a single Dirac cone in the SLAC formulation for eight Dirac components and the influence of dynamically induced long-range super-exchange interactions. The finite size behavior of dimensionless ratios and the finite size scaling properties of the Hubbard model with a single Dirac cone are shown to be superior compared to the honeycomb lattice. We extract the critical exponent believed to belong to the chiral Heisenberg Gross-Neveu-Yukawa universality class: The critical exponent ${ν= 1.02(3)}$ coincides for the two lattice types once honeycomb lattices of linear dimension ${L\ge 15}$ are considered. In contrast to the SLAC formulation, where the anomalous dimensions are estimated to be ${η_φ=0.73(1)}$ and ${η_ψ=0.09(1)}$, they remain less stable on honeycomb lattices, but tend towards the estimates from the SLAC formulation.

Figures (18)

• Figure 1: Comparison of estimates for the correlation-length exponent $\nu$, the boson anomalous dimension $\eta_{\phi}$ and the fermion anomalous dimension $\eta_{\psi}$, assuming the dynamic scaling exponent ${z=1}$, from analytical approaches Janssen14Zerf17Gracey18Knorr18Ladovrechis23TolosaSimeon25 and Monte Carlo simulations Assaad13Toldin15Otsuka16Tang18Buividovich18Buividovich19Liu19Otsuka20Ostmeyer20Liu21Ostmeyer21bXu21Otsuka22Yu23 for the ${N=8}$ chiral-Heisenberg universality class in chronological order. Gray vertical bars indicate the estimates and their errorbars from the SLAC-Hubbard model discussed in this manuscript.
• Figure 2: (a) Energy contours (light gray) and Berry flux pseudo spin map (blue arrows) for an $L=11$ lattice in momentum space. A closed contour around the Fermi point surface at the origin accumulates a flux of $\pi$ as can bee seen from the pseudo spin which sweeps twice the angle. The high symmetry path on a finite-size system is indicated by the red dashed line. The dispersion shown in (b) is discontinuous at the boundary of the Brillouin zone.
• Figure 3: The model shows two well defined phases: a gapless DSM regime where the order parameter ($m^2$) vanishes and a gapped, AFM ordered phase where the order parameter acquires finite values. Inset: the finite-size extrapolations of the single particle gap $\Delta_\text{sp}$ below, at and above the critical coupling ${U_c\approx 6}$.
• Figure 4: The single-particle spectral function ${A(\mathbf{k},\omega)}$ close to criticality for $L=17$ along the path in the Brillouin zone indicated in Fig. \ref{['fig:Berry']}. The non-interacting dispersion is indicated by the dashed line. The lowest excitation energy in the TDL are indicated for ${\mathbf{k}=M'}$, $X'$.
• Figure 5: Estimators for the renormalization of the Fermi velocity based on the finite-size energies at the smallest lattice momentum $\Delta(\delta\mathbf{k})$, the relativistic envelope of ${\Delta(\mathbf{k})}$ and the Dirac point $\Delta(\mathbf{0})$, . The finite-size show minor renormalization of the Fermi velocity.