Lattice Field Theory Analysis of the Chiral Heisenberg Model

Abstract: Motivated by ongoing interest in the universal behaviour of the Hubbard model of spinning electrons on honeycomb and $π$-flux lattices at the semi-metal -- Mott insulator phase transition, we formulate the \threeD~chiral Heisenberg model, a theory of relativistic fermions in three spacetime dimensions, as a lattice field theory using domain wall fermions. The contact interaction term preserves an SU(2) global symmetry. We perform numerical simulations using the Rational Hybrid Monte Carlo algorithm on system sizes $L^3\times L_s$ with $L\in{8,\ldots,24}$ and domain wall separation $L_s\in{8,16,24}$. We locate the phase transition corresponding to spontaneous SU(2)$\to$U(1) breaking, yielding critical exponent estimates $ν^{-1}=0.63(3)$, $ηΦ=1.42(8)$. These values are considerably removed from estimates obtained from simulations performed in (2+1)D, ie. with the time and spatial directions treated differently, but align more closely with analytic estimates obtained using 3D covariant field theory. We also present first results for the fermion correlator, ultimately needed for the determination of the exponent $ηΨ$, highlighting the need to rotate the fermion source to a common reference direction in isospace in order to obtain a signal.

• The paper demonstrates that using domain wall fermions preserves continuum chiral symmetry in a fully 3D covariant lattice formulation.
• It uses finite volume scaling with cubic fits to extract key critical exponents, reporting beta_c=0.465(11) and eta_phi=1.42(8).
• Results reveal significant discrepancies with (2+1)D studies, highlighting the impact of discretization strategy on universality classification.

Lattice Domain Wall Fermion Analysis of the 3D Chiral Heisenberg Model

Introduction and Motivation

The paper "Lattice Field Theory Analysis of the Chiral Heisenberg Model" (2603.29461) investigates the chiral Heisenberg universality class relevant to quantum criticality in correlated electron systems, such as the Hubbard model at half filling on honeycomb or $\pi$-flux lattices. These systems, in the weak coupling regime, are semi-metals with Dirac-like low-energy fermionic excitations. Upon tuning the on-site interaction $U$, a transition to a Mott insulator phase is observed, characterized by the spontaneous breaking of an SU(2) symmetry down to U(1). The effective field theory description is the chiral Heisenberg model in three dimensions (3D), i.e., the Gross-Neveu-Yukawa model with SU(2) symmetry and $N=2$ flavors of 4-component Dirac spinors coupled to a real bosonic triplet.

Traditional lattice studies typically distinguish between spatial and temporal directions, leading to (2+1)D simulations. The current work instead constructs a 3D covariant lattice field theory using domain wall fermions (DWF) to optimally preserve continuum symmetries, providing an alternative and systematically controlled approach for assessing universal critical behavior.

Model Formulation

The continuum Euclidean Lagrangian density for the chiral Heisenberg model is

$$\mathcal{L} = \bar\psi (\partial_\mu \gamma_\mu \otimes \mathbb{1} + g \vec\phi \cdot \mathbb{1} \otimes \vec\tau)\psi + \frac{1}{2} \vec\phi^2$$

with $\psi,\bar\psi$ representing $2$ flavors of 4-component Dirac spinors and $\vec\phi$ a real isotriplet scalar auxiliary field. The model possesses a global SU(2) flavor symmetry in the interaction term and reduces to U(2) invariance in the kinetic term.

For lattice regularization, DWF are deployed on a $L^3 \times L_s$ hypercubic lattice, achieving full 3D Lorentz symmetry as $L_s \to \infty$. The DWF approach is tailored to restore the Ginsparg-Wilson relations for Dirac fermions, thus suppressing explicit breaking of continuum chiral symmetry. The fermion-boson interaction is defined only on the domain walls, omitting the need for a bosonic Pauli-Villars kernel. The auxiliary $\vec\phi$ fields are simulated via the Rational Hybrid Monte Carlo (RHMC) algorithm.

Numerical Results: Phase Structure and Critical Exponents

Simulations span system sizes $U$0 with $U$1 and domain wall separations $U$2. The order parameter is the volume-averaged magnitude $U$3 of the bosonic field. Due to absence of explicit symmetry breaking (e.g., no bare fermion mass), $U$4 drifts across the SO(3) vacuum manifold in simulations, necessitating use of the pseudo-order parameter $U$5.

The phase transition associated with SU(2) $U$6 U(1) breaking is located by analysis of the Binder cumulant and subsequently pinned down with finite volume scaling (FVS) collapses. The FVS ansatz adopted is

$U$7

with $U$8. Quadratic and cubic polynomial fits confirm robust extraction of critical exponents with negligible dependence on $U$9, indicating early restoration of continuum symmetries in this model compared to, e.g., the 3D Thirring case. The final reported exponents from cubic FVS fits at $N=2$0 are:

$N=2$1

Utilizing hyperscaling, this yields $N=2$2.

Comparison with Previous Work

A systematic compilation of critical exponent estimates from alternative simulation methodologies as well as analytic treatments (e.g., $N=2$3-expansions, FRG, large-$N=2$4) shows that the present results for $N=2$5 and $N=2$6 are outliers. Previous (2+1)D QMC and HMC studies cluster around $N=2$7 and $N=2$8, subject to strong finite size effects and significant spread. 3D covariant approaches, including this work, show consistently smaller $N=2$9 and larger $$\mathcal{L} = \bar\psi (\partial_\mu \gamma_\mu \otimes \mathbb{1} + g \vec\phi \cdot \mathbb{1} \otimes \vec\tau)\psi + \frac{1}{2} \vec\phi^2$$0. A pronounced anti-correlation between $$\mathcal{L} = \bar\psi (\partial_\mu \gamma_\mu \otimes \mathbb{1} + g \vec\phi \cdot \mathbb{1} \otimes \vec\tau)\psi + \frac{1}{2} \vec\phi^2$$1 and $$\mathcal{L} = \bar\psi (\partial_\mu \gamma_\mu \otimes \mathbb{1} + g \vec\phi \cdot \mathbb{1} \otimes \vec\tau)\psi + \frac{1}{2} \vec\phi^2$$2 is observed across the literature. The results from this DWF study populate the extremal edge of the "3D" cluster, emphasizing the impact of full Lorentz-covariant discretization.

Fermion Correlator Analysis

The fermion two-point function is reconstructed from DWF-propagator components projected onto the domain walls, yielding time-slice correlators $$\mathcal{L} = \bar\psi (\partial_\mu \gamma_\mu \otimes \mathbb{1} + g \vec\phi \cdot \mathbb{1} \otimes \vec\tau)\psi + \frac{1}{2} \vec\phi^2$$3 and $$\mathcal{L} = \bar\psi (\partial_\mu \gamma_\mu \otimes \mathbb{1} + g \vec\phi \cdot \mathbb{1} \otimes \vec\tau)\psi + \frac{1}{2} \vec\phi^2$$4. To mitigate the vacuum manifold drift during simulations, each measurement is rotated such that the source aligns along a reference direction in isospace. The Truncated Hankel Correlator (THC) method is used to extract spectral information from matrix-valued correlators.

Key numerical findings in this sector include:

• The $$\mathcal{L} = \bar\psi (\partial_\mu \gamma_\mu \otimes \mathbb{1} + g \vec\phi \cdot \mathbb{1} \otimes \vec\tau)\psi + \frac{1}{2} \vec\phi^2$$5 correlator, after rotation, indicates that only the component along the symmetry-breaking direction retains significant signal. Off-diagonal entries vanish within statistical uncertainty, confirming the residual U(1) symmetry.
• The ground state energy from the fermion correlator is largely independent of the coupling $$\mathcal{L} = \bar\psi (\partial_\mu \gamma_\mu \otimes \mathbb{1} + g \vec\phi \cdot \mathbb{1} \otimes \vec\tau)\psi + \frac{1}{2} \vec\phi^2$$6, showing that fermions remain gapped across the phase transition. Amplitude analysis suggests decreasing interaction with the bosonic field as $$\mathcal{L} = \bar\psi (\partial_\mu \gamma_\mu \otimes \mathbb{1} + g \vec\phi \cdot \mathbb{1} \otimes \vec\tau)\psi + \frac{1}{2} \vec\phi^2$$7 increases, but no clear critical scaling signal is seen in the correlator itself within current lattice volumes.
• Systematic finite size effects and the vanishing of the correlator amplitude for large $$\mathcal{L} = \bar\psi (\partial_\mu \gamma_\mu \otimes \mathbb{1} + g \vec\phi \cdot \mathbb{1} \otimes \vec\tau)\psi + \frac{1}{2} \vec\phi^2$$8 challenge robust extraction of the fermionic critical exponent $$\mathcal{L} = \bar\psi (\partial_\mu \gamma_\mu \otimes \mathbb{1} + g \vec\phi \cdot \mathbb{1} \otimes \vec\tau)\psi + \frac{1}{2} \vec\phi^2$$9.

Implications and Outlook

This analysis underscores the strong influence of preserving continuum symmetries in numerical studies of quantum criticality. The pronounced discrepancy between 3D covariant and (2+1)D non-covariant simulations for critical exponents highlights the necessity of methodologically controlled studies when characterizing universality classes. The results call for critical reevaluation of systematic errors, including FVS ansatz and lattice artifacts, as well as the need for larger volumes and more direct comparisons to emergent symmetry scenarios or conformal bootstrap predictions.

Practically, the paper establishes DWF-based 3D lattice field theorization as a viable route for non-perturbative studies of Dirac-critical transitions with dynamically realized continuous symmetry breaking. Theoretically, it raises important questions regarding the proper assignment of universality classes to electronic model quantum transitions, and the role of symmetry realization and discretization strategy on scaling properties.

Future directions include extension to larger lattices to better resolve the fermion spectrum and critical scaling in the correlator sector, direct computation of $\psi,\bar\psi$0, and cross-comparison with independent approaches such as conformal bootstrap or tensor network techniques. The observed anti-correlation of estimated exponents indicates a methodological challenge in decoupling different scaling dimensions, suggesting new statistical or algorithmic innovations may be required for further progress.

Conclusion

This work provides a systematic non-perturbative lattice study of the 3D chiral Heisenberg model using DWF, yielding critical exponents that diverge from prior (2+1)D numerical investigations and align more with analytic covariant field theory expectations. The implications for the interpretation of quantum critical phenomena in strongly correlated Dirac systems are substantial, directing attention to both practical simulation strategies and theoretical classification schemes within quantum many-body and field-theoretic settings. A comprehensive understanding of universality in these systems will require synthesis of insights from both covariant and non-covariant approaches, as well as continued refinement of statistical and lattice field theory methodologies.