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

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.

💡 Research Summary

The manuscript presents a comprehensive quantum Monte‑Carlo (QMC) investigation of the chiral Heisenberg Gross‑Neveu‑Yukawa (GNY) quantum critical point in (2+1) dimensions, focusing on the case of N = 8 Dirac components (four orbitals × two spin flavors). The authors compare two lattice realizations that host a single Dirac cone: (i) a conventional honeycomb lattice with a π‑flux configuration, and (ii) a lattice built from SLAC fermions, i.e., a non‑local hopping construction that reproduces the exact linear Dirac dispersion ε(k)=±v_F|k| for every momentum point in the Brillouin zone.

The SLAC formulation is introduced on a square lattice of linear size L = 2N_k (even) or L = 2N_k+1 (odd). The hopping amplitudes decay as t(r)∝1/r (odd L) or contain a constant term for even L, leading to a perfectly relativistic spectrum with a 2N_f‑fold degeneracy at the Dirac point. The authors emphasize that the Berry flux of π associated with the Dirac cone is correctly reproduced, and that the non‑local hopping only couples sites along the principal axes, thus avoiding a truly all‑to‑all interaction.

The interacting model adds a Hubbard repulsion U that preserves SU(2) spin symmetry but drives antiferromagnetic (AF) order at strong coupling, thereby realizing the chiral Heisenberg transition. The QMC algorithm is a projector auxiliary‑field determinant scheme free of the sign problem. Technical parameters include a projection length 2Θ = 70, imaginary‑time step Δτ = 0.1, and measurement after τ = 30. The computational cost scales as O(64 L⁶) for the SLAC case (four orbitals per site) versus O(8 L⁶) for the honeycomb lattice. Despite the higher nominal cost, the SLAC approach exhibits dramatically reduced finite‑size corrections because the linear dispersion is exact for all L, allowing accurate extraction of critical exponents from relatively modest system sizes.

Finite‑size scaling (FSS) analyses are performed using dimensionless ratios such as the correlation‑length ratio ξ/L and the Binder cumulant. By collapsing data across several lattice sizes, the authors obtain a correlation‑length exponent ν = 1.02(3), which is consistent with previous analytical estimates (ε‑expansion, functional RG) and with conformal bootstrap results for the same universality class.

Anomalous dimensions are extracted from the scaling of the staggered magnetization (bosonic field) and the fermionic Green’s function. In the SLAC implementation the boson anomalous dimension is η_ϕ = 0.73(1) and the fermion anomalous dimension is η_ψ = 0.09(1). These values are remarkably stable across system sizes. For the honeycomb lattice, the same exponents show a strong size dependence; only for L ≥ 15 do they begin to approach the SLAC estimates, but the statistical uncertainties remain larger. This demonstrates that the honeycomb geometry suffers from pronounced non‑linearities in the Dirac spectrum at small L, which inflate scaling corrections.

A notable discussion concerns the “dynamically induced” long‑range antiferromagnetic exchange J ∼ t²/U ∼ 1/(U r²) that arises from the 1/r hopping of SLAC fermions. The authors argue that such a power‑law exchange with exponent α = 2 does not satisfy the conditions of the Mermin‑Wagner theorem in two dimensions, because the energy cost of a disorder droplet scales as L^{3‑α}=L, which grows with system size. Consequently, spontaneous breaking of a continuous symmetry can occur even at finite temperature in the SLAC‑Hubbard model, without violating the theorem’s assumptions. However, at zero temperature (the regime of interest for the GNY critical point) the theorem is irrelevant, and the long‑range exchange merely reduces finite‑size effects rather than altering the universality class.

The paper concludes that the SLAC fermion construction provides a superior platform for studying Dirac‑fermion quantum criticality. It yields precise critical exponents with modest lattice sizes, while the honeycomb lattice requires substantially larger systems to achieve comparable accuracy. The authors acknowledge that the non‑local nature of SLAC hopping may pose challenges for extensions involving dynamical gauge fields or more intricate interaction structures, but they argue that for purely fermionic Hubbard‑type models the approach is both reliable and efficient.

Overall, the work clarifies why previous QMC studies of the chiral Heisenberg GNY transition have reported disparate exponent values: the dominant source of discrepancy is finite‑size scaling on conventional lattices. By employing a lattice that preserves the continuum Dirac dispersion exactly, the authors obtain a coherent set of critical exponents—ν ≈ 1.02, η_ϕ ≈ 0.73, η_ψ ≈ 0.09—thereby establishing a benchmark for future numerical and analytical investigations of interacting Dirac systems.