Gross-Neveu-Heisenberg Universality Class

• Gross-Neveu-Heisenberg universality class is defined by a continuous quantum phase transition where gapless Dirac fermions couple to a three-component order parameter, leading to spontaneous SU(2) symmetry breaking.
• Its structure is analyzed via continuum renormalization group methods, large-N expansions, and sign-problem-free quantum Monte Carlo across diverse lattice realizations.
• Lattice studies on honeycomb, square, and SLAC geometries confirm universal critical behavior despite variations in Dirac cone multiplicity and finite-size scaling corrections.

The Gross-Neveu-Heisenberg universality class, also called the chiral Heisenberg Gross-Neveu or chiral Heisenberg Gross-Neveu-Yukawa class, is the fermionic universality class in $2+1$ dimensions that describes a continuous quantum phase transition between a Dirac semimetal and an antiferromagnetic insulator in which $SU(2)$ spin rotational symmetry is spontaneously broken. Its defining structure is a set of gapless Dirac fermions coupled to a three-component order parameter, and its modern characterization combines continuum RG, large-$N$ critical-point methods, and sign-problem-free quantum Monte Carlo across several inequivalent lattice realizations (Lang et al., 19 Mar 2025).

1. Definition and symmetry structure

In its standard form, the universality class is the Heisenberg, or $O(3)$, member of the Gross-Neveu family: the order parameter is a three-component vector, $N_b=3$, and the symmetry-breaking pattern is the onset of antiferromagnetic order from a gapless Dirac phase (Zhou et al., 2020). The same class is therefore referred to as “chiral Heisenberg” when emphasis is placed on the gapless Dirac fermions, and as Gross-Neveu-Heisenberg when emphasis is placed on the symmetry of the order parameter and the underlying four-fermion interaction (Gracey, 2018).

A standard bosonized low-energy description is the Gross-Neveu-Yukawa form

$$\mathcal{L}_\text{GNYH} = \bar\psi_a (\gamma_\mu \partial_\mu ) \psi_a + \frac12 (\partial_\mu \phi)^2 - h \phi_\alpha \bar\psi_a \sigma_\alpha \psi_a + \lambda (\phi_\alpha \phi_\alpha)^2,$$

with a real three-component field $\phi_\alpha$ coupled to Dirac fermions through a Yukawa term (Ray et al., 2021). The equivalent Gross-Neveu formulation instead uses a four-fermion interaction and is related to the Yukawa theory by bosonization with an auxiliary field; this equivalence underlies the interchangeable use of Gross-Neveu and Gross-Neveu-Yukawa language in the literature (Gracey, 2018).

A central feature is emergent Lorentz symmetry at criticality. In a bilayer-graphene realization built from a nonrelativistic quadratic band touching, the nematic-to-coexistence transition flows to a relativistic fixed point with dynamical exponent $z=1$, despite the microscopic anisotropy and the absence of exact continuous rotational symmetry in the lattice model (Ray et al., 2021). This supports the broader interpretation of the class as a relativistic infrared fixed point of interacting Dirac matter coupled to an $O(3)$ order parameter.

2. Continuum formulations and analytical approaches

The analytical study of the universality class has proceeded along three main directions: large-$N$ critical-point methods, the $SU(2)$0 expansion of the Yukawa theory, and the $SU(2)$1 expansion of the fermionic theory. In the large-$SU(2)$2 formalism, the critical exponents are expanded as

$SU(2)$3

with explicit all-dimensional expressions obtained for $SU(2)$4 to $SU(2)$5 and for $SU(2)$6 and $SU(2)$7 to $SU(2)$8 by combining skeleton Schwinger-Dyson equations and the large-$SU(2)$9 conformal bootstrap (Gracey, 2018). For $N$0 in $N$1, Padé-resummed large-$N$2 estimates give $N$3, $N$4, and $N$5, illustrating both the reach and the limitations of the expansion at physically modest flavor number (Gracey, 2018).

The lower-critical-dimension expansion around $N$6 exposed a structural feature that is specific to the Heisenberg case: a single four-fermion coupling is not sufficient. A Fierz-complete treatment requires six independent interaction channels,

$N$7

and the resulting channel competition materially affects the fixed-point structure and the correction-to-scaling spectrum (Ladovrechis et al., 2022). After interpolation between the $N$8 and $N$9 expansions, the estimates quoted for the single-layer-graphene case are

$O(3)$0

together with an unusually small leading correction-to-scaling exponent, $O(3)$1 (Ladovrechis et al., 2022). The small $O(3)$2 is significant because it implies slow RG flow into the asymptotic regime.

The $O(3)$3 framework remains the standard perturbative language for Yukawa formulations and for gauged extensions. In the ungauged case it provides the upper-critical-dimension anchor for interpolation; in the gauged case it produces a distinct chiral Heisenberg QED$O(3)$4-GNY fixed point, not the Gross-Neveu-Heisenberg fixed point itself (Zhou et al., 2020).

3. Lattice realizations and numerical evidence

The canonical condensed-matter realization is the half-filled Hubbard model on the honeycomb lattice, where the transition separates a Dirac semimetal from an antiferromagnetic insulator. More recent work established that the same criticality also appears in microscopically different settings, including a square-lattice Hubbard model with a $O(3)$5-wave pairing field and a SLAC-fermion implementation with a single Dirac cone, provided the low-energy symmetry and Dirac content are the same (Otsuka et al., 2021).

In the square-lattice $O(3)$6-wave model, the pairing field induces four Dirac nodes in the noninteracting spectrum at half filling, and the continuum limit remains the chiral Heisenberg Gross-Neveu theory with $O(3)$7 once spin is included (Otsuka et al., 2020). Auxiliary-field QMC studies found that the square-lattice model and the honeycomb Hubbard model have the same quantum criticality despite different unit-cell structures, different numbers of Dirac cones in the lattice Brillouin zone, and tunable Dirac-cone anisotropy; the anisotropy and the way of counting the fermion components do not change the critical exponents within numerical accuracy (Otsuka et al., 2021).

A 2025 large-scale QMC study sharpened this picture by comparing honeycomb and SLAC regularizations directly near the critical point (Lang et al., 19 Mar 2025). In that work, the SLAC construction realizes a perfect single Dirac cone across the Brillouin zone for eight Dirac components and generates dynamically induced long-range super-exchange interactions, $O(3)$8, through its long-range hopping $O(3)$9. The critical behavior, however, remains unchanged: the long-range super-exchange does not alter the universality class, the Dirac semimetal persists up to $N_b=3$0, and the antiferromagnetic phase exhibits gapless Goldstone modes (Lang et al., 19 Mar 2025). The same study further reported that the finite-size behavior of dimensionless ratios and the finite-size scaling properties are superior in the SLAC formulation relative to the honeycomb lattice, making the SLAC regularization a numerically cleaner route to the asymptotic scaling regime (Lang et al., 19 Mar 2025).

4. Critical exponents, finite-size scaling, and corrections to scaling

The extraction of critical data in lattice realizations is based on RG-invariant correlation ratios and standard finite-size scaling. In the SLAC-versus-honeycomb comparison, a representative correlation ratio is

$N_b=3$1

and the scaling analysis uses

$N_b=3$2

together with

$N_b=3$3

to locate $N_b=3$4 and fit $N_b=3$5, $N_b=3$6, and $N_b=3$7 (Lang et al., 19 Mar 2025).

Different methods and realizations quote different exponent sets, partly because they address different flavor numbers and partly because finite-size corrections remain nontrivial. The following values are among the most frequently cited benchmarks:

Setting	Flavor / component count	Reported exponents
Honeycomb vs. SLAC Hubbard (Lang et al., 19 Mar 2025)	$N_b=3$8 Dirac components	$N_b=3$9, $$\mathcal{L}_\text{GNYH} = \bar\psi_a (\gamma_\mu \partial_\mu ) \psi_a + \frac12 (\partial_\mu \phi)^2 - h \phi_\alpha \bar\psi_a \sigma_\alpha \psi_a + \lambda (\phi_\alpha \phi_\alpha)^2,$$0, $$\mathcal{L}_\text{GNYH} = \bar\psi_a (\gamma_\mu \partial_\mu ) \psi_a + \frac12 (\partial_\mu \phi)^2 - h \phi_\alpha \bar\psi_a \sigma_\alpha \psi_a + \lambda (\phi_\alpha \phi_\alpha)^2,$$1
Square-lattice Hubbard with $$\mathcal{L}_\text{GNYH} = \bar\psi_a (\gamma_\mu \partial_\mu ) \psi_a + \frac12 (\partial_\mu \phi)^2 - h \phi_\alpha \bar\psi_a \sigma_\alpha \psi_a + \lambda (\phi_\alpha \phi_\alpha)^2,$$2-wave pairing (Otsuka et al., 2020)	$$\mathcal{L}_\text{GNYH} = \bar\psi_a (\gamma_\mu \partial_\mu ) \psi_a + \frac12 (\partial_\mu \phi)^2 - h \phi_\alpha \bar\psi_a \sigma_\alpha \psi_a + \lambda (\phi_\alpha \phi_\alpha)^2,$$3	$$\mathcal{L}_\text{GNYH} = \bar\psi_a (\gamma_\mu \partial_\mu ) \psi_a + \frac12 (\partial_\mu \phi)^2 - h \phi_\alpha \bar\psi_a \sigma_\alpha \psi_a + \lambda (\phi_\alpha \phi_\alpha)^2,$$4, $$\mathcal{L}_\text{GNYH} = \bar\psi_a (\gamma_\mu \partial_\mu ) \psi_a + \frac12 (\partial_\mu \phi)^2 - h \phi_\alpha \bar\psi_a \sigma_\alpha \psi_a + \lambda (\phi_\alpha \phi_\alpha)^2,$$5, $$\mathcal{L}_\text{GNYH} = \bar\psi_a (\gamma_\mu \partial_\mu ) \psi_a + \frac12 (\partial_\mu \phi)^2 - h \phi_\alpha \bar\psi_a \sigma_\alpha \psi_a + \lambda (\phi_\alpha \phi_\alpha)^2,$$6
Interpolated single-layer-graphene case (Ladovrechis et al., 2022)	$$\mathcal{L}_\text{GNYH} = \bar\psi_a (\gamma_\mu \partial_\mu ) \psi_a + \frac12 (\partial_\mu \phi)^2 - h \phi_\alpha \bar\psi_a \sigma_\alpha \psi_a + \lambda (\phi_\alpha \phi_\alpha)^2,$$7	$$\mathcal{L}_\text{GNYH} = \bar\psi_a (\gamma_\mu \partial_\mu ) \psi_a + \frac12 (\partial_\mu \phi)^2 - h \phi_\alpha \bar\psi_a \sigma_\alpha \psi_a + \lambda (\phi_\alpha \phi_\alpha)^2,$$8, $$\mathcal{L}_\text{GNYH} = \bar\psi_a (\gamma_\mu \partial_\mu ) \psi_a + \frac12 (\partial_\mu \phi)^2 - h \phi_\alpha \bar\psi_a \sigma_\alpha \psi_a + \lambda (\phi_\alpha \phi_\alpha)^2,$$9, $\phi_\alpha$0
Bilayer-graphene nematic-to-coexistence transition (Ray et al., 2021)	$\phi_\alpha$1	$\phi_\alpha$2, $\phi_\alpha$3, $\phi_\alpha$4

For the $\phi_\alpha$5 lattice problem, the 2025 QMC study found that $\phi_\alpha$6 coincides for SLAC and honeycomb once honeycomb lattices of linear dimension $\phi_\alpha$7 are included, while the anomalous dimensions remain less stable on the honeycomb lattice but tend toward the SLAC estimates (Lang et al., 19 Mar 2025). Earlier square-lattice AFQMC work obtained a larger $\phi_\alpha$8, $\phi_\alpha$9, together with $z=1$0 and $z=1$1, and interpreted the agreement with honeycomb data as a confirmation of universality across lattice geometries (Otsuka et al., 2020).

The persistence of spread in quoted exponents is therefore not usually interpreted as evidence against universality. Rather, the $z=1$2 analysis suggests that the leading correction-to-scaling exponent can be unusually small, especially for the single-layer-graphene case, so finite-size extrapolations may require explicit correction terms and very large systems before the asymptotic regime is cleanly visible (Ladovrechis et al., 2022). This suggests that part of the long-standing numerical scatter, especially in $z=1$3 and $z=1$4, is a corrections-to-scaling problem rather than a mismatch of universality classes.

5. Generalizations and nearby fixed points

The Gross-Neveu-Heisenberg fixed point also appears in systems that do not begin from a monolayer Dirac semimetal. In Bernal-stacked bilayer graphene, an effective model with competing nematic and layer-antiferromagnetic orders exhibits a continuous quantum phase transition from a nematic phase to a coexistence phase, and that transition falls into the $z=1$5-dimensional relativistic Gross-Neveu-Heisenberg universality class (Ray et al., 2021). At the critical point, four massless Dirac cones emerge from the nematic background, the antiferromagnetic order parameter acts as the mass-generating field, and Lorentz-breaking perturbations are RG-irrelevant; by contrast, the coexistence-to-antiferromagnetic transition is weakly first order because the honeycomb bilayer lacks continuous spatial rotational symmetry (Ray et al., 2021).

A nearby but distinct class arises when the Dirac fermions couple to a dynamical $z=1$6 gauge field. In the chiral Heisenberg QED$z=1$7-GNY theory, the order parameter is still a three-component vector, but the gauge field modifies the fixed point and changes the critical exponents (Zhou et al., 2020). At one loop in $z=1$8, the boson anomalous dimension and inverse correlation-length exponent are

$z=1$9

$O(3)$0

with Padé estimates reported for $O(3)$1 and $O(3)$2 (Zhou et al., 2020). The same work emphasizes that these gauged exponents differ substantially from deconfined-critical-point numerics, especially in $O(3)$3, which indicates that higher-loop or nonperturbative effects are needed before any strict quantitative identification can be made (Zhou et al., 2020).

The distinction between ungauged and gauged fermionic criticality is also evident from the QED$O(3)$4-Gross-Neveu-XY literature: the presence of a dynamical gauge field produces new fixed points and new scaling dimensions that are not obtained by simply adding together ungauged Gross-Neveu and QED$O(3)$5 physics (Janssen et al., 2020). By the same logic, the Gross-Neveu-Heisenberg universality class is not merely “Dirac fermions plus antiferromagnetism,” but the specific ungauged $O(3)$6 fixed point selected by its Yukawa or four-fermion structure.

6. Boundary criticality, dynamical properties, and current outlook

Recent work has extended the Gross-Neveu-Yukawa framework to boundary criticality. For interacting Dirac fermions on a honeycomb lattice with armchair boundaries, mean-field theory identifies ordinary, special, and extraordinary boundary transitions at the bulk quantum critical point, while a $O(3)$7 RG treatment computes boundary critical exponents for the GNY family and explicitly tabulates the chiral Heisenberg case (Jiang et al., 17 Mar 2025). The fermions obey a Dirichlet boundary condition in all cases, while the bosonic order parameter obeys Dirichlet at the ordinary transition and Neumann at the special transition; for the chiral Heisenberg class, the leading boundary fermion dimensions are

$O(3)$8

with corresponding formulas for $O(3)$9, $N$0, and $N$1 also given (Jiang et al., 17 Mar 2025). This places the class within the broader BCFT taxonomy of ordinary, special, and extraordinary transitions.

The universality class also admits dynamical probes beyond static critical exponents. In a large-$N$2 study of many-body quantum chaos at the Gross-Neveu critical point, the Lyapunov exponent was found to scale as

$N$3

and the low-energy quantum scattering rate as

$N$4

That analysis focused on the $N$5 Gross-Neveu transition, but it further argued that chiral XY and chiral Heisenberg universality classes have the same leading-order behavior for $N$6 because the relevant polarization operator structure is dominated by fermionic loops (Jian et al., 2018). This suggests that the Gross-Neveu-Heisenberg class should be viewed not only as a static universality class of order-parameter onset, but also as a dynamical regime of rapid scrambling and ill-defined quasiparticles at finite temperature.

Two broad issues remain central. First, analytical and numerical exponent estimates are not yet fully reconciled across all flavor numbers and regularizations. Second, the class sits inside a crowded neighborhood of related fixed points—gauged, anisotropic, easy-plane, and boundary variants—so terminology can obscure rather than clarify. The most robust statement supported across modern work is that the Gross-Neveu-Heisenberg universality class is the ungauged relativistic $N$7 fermion-boson fixed point governing Dirac-semimetal to antiferromagnetic-insulator criticality, and that its universal content survives substantial changes in lattice geometry, cone multiplicity, anisotropy, and even certain dynamically generated long-range interactions, provided the symmetry, dimensionality, and low-energy Dirac structure are preserved (Lang et al., 19 Mar 2025).