Gross-Neveu-Yukawa Model Overview

• The Gross-Neveu-Yukawa model is a quantum field theory where Dirac fermions interact with bosonic order parameter fields via a Yukawa coupling, providing a framework to analyze critical phenomena.
• It employs renormalization group techniques and high-loop epsilon expansions to precisely calculate critical exponents and tackle symmetry-breaking and multicriticality in interacting Dirac systems.
• The model underpins studies on transitions in graphene, Weyl semimetals, and topological materials, and reveals insights into phenomena like emergent supersymmetry and deconfined quantum criticality.

The Gross-Neveu-Yukawa (GNY) model is a class of (typically Euclidean) quantum field theories in which Dirac fermions are coupled via a Yukawa interaction to one or more bosonic order parameter fields, usually subject to symmetry constraints. In high-energy and condensed matter physics, the GNY model systematically unifies the analysis of universal critical phenomena in interacting Dirac systems, including metal-insulator transitions, deconfined quantum criticality, emergent supersymmetry, and quantum phase transitions in semimetals (such as graphene and Weyl systems). The model serves as the ultraviolet (UV) complete counterpart to the original purely fermionic Gross-Neveu model by allowing renormalizability in 4 – ε dimensions and precise control over symmetry-breaking, operator content, and quantum corrections.

1. Model Formulation and Core Lagrangian Structure

The canonical GNY Lagrangian in $d=4-\epsilon$ Euclidean spacetime dimensions reads: $$\mathcal{L} = \overline{\psi}(\partial\!\!\!/ + g \phi)\psi + \frac{1}{2} \phi(-\partial^2 + m^2)\phi + \frac{\lambda}{4!} \phi^4$$ where $\psi$ is an $N$-component (Dirac, Majorana, or two-component) fermion field depending on the symmetry class, $\phi$ is a real scalar (for Ising-type), O($M$) vector, or higher-rank tensor bosonic field, and $g$, $\lambda$ are the Yukawa and self-interaction couplings.

• For chiral Ising, $\phi$ is a real singlet and $\psi$ is typically a Dirac (or Majorana) fermion.
• For chiral Heisenberg or XY, $\phi$ transforms under SU(2) or U(1), and the form of Yukawa and quartic invariants adapts accordingly.
• For matrix or tensorial GNY models, e.g., with SO(2N)→SO(N)×SO(N) order (Han et al., 3 Jun 2024), $\phi$ is a traceless symmetric matrix, and the quartic potential involves multiple invariant couplings.

The local four-fermion interaction of the original Gross-Neveu model is linearized through a Hubbard–Stratonovich transformation, mapping it onto Yukawa interaction with the auxiliary $\phi$ field.

2. Symmetry Structure, Universality Classes, and Multicriticality

The symmetry class and operator content of a GNY theory are governed by the representation of the bosonic order parameter and the corresponding fermion bilinear. Possible cases include:

Universality Class	Order Parameter $\phi$	Fermionic Bilinear	Symmetry
Chiral Ising (Z₂)	Real scalar	$\overline{\psi} \psi$	Z₂
Chiral XY (U(1))	Complex scalar	$\overline{\psi} (\tau^+)\psi$	U(1)
Chiral Heisenberg (SU(2))	3-component vector	$\overline{\psi} (\sigma^a) \psi$	SU(2)
Matrix (SO(2N))	Symmetric traceless	Multi-component	SO(2N)

In multicritical settings, such as graphene with competing spin singlet and triplet instabilities, the GNY framework reveals that disparate orders can meet at a single Gross-Neveu-type critical point (Roy, 2011). The explicit breaking of spin or lattice symmetries (e.g., Z₂ × O(2) anisotropy) typically becomes RG-irrelevant near the critical point, which is characterized by emergent restoration of symmetry (e.g., to SO(3)).

Physical symmetry-breaking perturbations can arise from microscopic sources such as spin-orbit coupling, lattice deformations, or external fields. In the RG sense, terms that reduce the order parameter symmetry (e.g., selecting a preferred spin direction via external Zeeman fields) are found to be irrelevant: their coupling flows to zero at long distances, and thus they do not affect the universality class or leading critical behavior (Roy, 2011).

3. Renormalization Group and Critical Exponents

Beta Functions and Higher-Loop Calculations

The GNY model is perturbatively renormalizable in 4 – ε dimensions. Beta functions for the rescaled Yukawa coupling $y=g^2$ and quartic $X$ (proportional to $\lambda$) take the schematic form: $\beta_y = dy/d\ln\mu, \quad \beta_X = dX/d\ln\mu$ Critical exponents are extracted at the non-Gaussian (Wilson-Fisher) fixed point where $\beta_y=\beta_X=0$.

• Low-order (one-loop) exponents (Roy, 2011):

$$\nu = \frac{1}{2} + \frac{21}{55}\epsilon;\quad \eta_b = \frac{4}{5}\epsilon;\quad \eta_\psi = \frac{3}{10}\epsilon$$

• Calculations have reached three-loop (Mihaila et al., 2017), four-loop (Zerf et al., 2017), and five-loop (Gracey et al., 30 Jul 2025) order, yielding high-precision estimates for $\nu$, $\eta_\psi$, $\eta_\phi$, and correction-to-scaling exponents $\omega_\pm$.
• The stability and universality properties, including subleading corrections, are controlled by the eigenvalues of the stability (Hessian) matrix of the beta functions evaluated at the fixed point (Gracey, 2017, Manashov et al., 2017).

Padé and Borel–Padé resummation schemes are used to extrapolate high-loop $\epsilon$-expansions to $d=3$ (Zerf et al., 2017, Gracey et al., 30 Jul 2025). For $N=2$ (graphene), recent five-loop analyses yield: $$\begin{align*} \eta_\psi &\approx (1/14)\,\epsilon - (71/10584)\,\epsilon^2 + \cdots\ \eta_\phi &\approx (4/7)\,\epsilon + (109/882)\,\epsilon^2 + \cdots\ 1/\nu &\approx 2 - (20/21)\,\epsilon + \cdots\ \end{align*}$$ The critical exponents are essential for characterizing the universality class and can be compared directly with Monte Carlo (Lang et al., 2018, Liu et al., 2019), FRG, and CFT bootstrap calculations (Erramilli et al., 2022, Mitchell et al., 18 Jun 2024).

Emergent Supersymmetry and Scaling Relations

For certain fractional fermion flavors, the GNY model exhibits emergent supersymmetry (SUSY) at criticality; e.g., for $N=1/4$ (chiral Ising) or $N=1/2$ (chiral XY) (Zerf et al., 2017). In these cases, superscaling relations such as

$$\nu^{-1} = \frac{D-\eta}{2}, \quad \eta_\phi = \eta_\psi = \eta$$

are satisfied to all loop orders, contingent on the implementation of suitable dimensional regularization (DREG₃) that captures the algebra of $\gamma$-matrices in three dimensions.

4. Physical Applications: Dirac and Weyl Systems, Quantum Phase Transitions

The GNY universality class is central to the description of a broad range of quantum transitions in Dirac and Weyl semimetals:

• Graphene: The transition from a Dirac semimetal to a charge-density-wave (CDW) or spin-density-wave (SDW) insulator is described by the chiral Ising or Heisenberg GNY universality class ($N=2$) (Roy, 2011, Mihaila et al., 2017, Zerf et al., 2017).
• Spinless fermions on the honeycomb lattice ($N=1$): the corresponding metal-insulator transition is captured by the $N=1$ GNY exponents (Mihaila et al., 2017, Liu et al., 2019).
• Surface transitions in topological crystalline insulators, or emergent SUSY at the surface of topological superconductors, are naturally described in the $N=1/4$ (chiral Ising) or $N=1/2$ (chiral XY) universality classes (Zerf et al., 2017).
• Deconfined criticality in square lattice antiferromagnets (Néel–to–VBS transitions): the QED₃–GNY model is conjectured to be dual to the deconfined transition, with enlarged SO(5) symmetry and critical exponents supported by 1/N expansion and numerical resummations (Boyack et al., 2018).
• Weyl semimetals with disorder: the replica limit $N\to0$ of the GNY theory describes the transition from a Weyl semimetal to a diffusive metal (Mihaila et al., 2017).

Experimental consequences include the scaling of transport coefficients, spectral functions, and boundary critical behavior in designer Dirac systems (Jiang et al., 17 Mar 2025).

5. Boundary Criticality and Operator Content

Boundary conditions play a crucial role in systems with edges or interfaces:

• For Dirac fermions, the natural boundary condition at an armchair edge is (Jiang et al., 17 Mar 2025):

$-\gamma^1 \psi|_{\text{bdy}} = \psi|_{\text{bdy}}$

• The bosonic field may obey Dirichlet or Neumann boundary conditions, corresponding respectively to the ordinary and special transitions in boundary criticality.
• Critical exponents for boundary operators are computed via one-loop $\epsilon$-expansion, with scaling dimensions for the derivative of the boundary field (Dirichlet) or the field itself (Neumann) determined by anomalous dimensions $Z_{\partial\hat{\phi}}$ and $Z_{\hat{\psi}}$.

The interplay between bulk and boundary criticality leads to a classification of possible transitions: ordinary, extraordinary, and special, with direct experimental relevance for scanning tunneling microscopy and boundary-sensitive observables.

6. Tensorial Extensions and Higher-Order (Matrix) GNY Models

Recent developments include matrix/tensorial generalizations where the order parameter is a traceless symmetric tensor under SO(N), coupled to Majorana fermions (Han et al., 3 Jun 2024, Han et al., 25 Jun 2025). Key features include:

• Multiple critical flavor numbers $N_{f,c1}$, $N_{f,c2}$ separating regimes of first-order and continuous transitions.
• For $N_f < N_{f,c1}$, no fixed point exists; for $N_{f,c1}<N_f<N_{f,c2}$, a "real" fixed point emerges in an unstable region of coupling space; only for $N_f > N_{f,c2}$ is a stable critical fixed point found.
• Loop corrections substantially reduce $N_{f,c2}$ compared to one-loop, affecting the accessibility of the continuous transition for systems such as graphene.
• Critical exponents in the stable regime (large $N_f$) are computed via the $\epsilon$-expansion, with explicit dependence on $N$, $N_f$, and quartic coupling ratios.

Matrix GNY models provide a unifying language for multi-component mass orders in correlated Dirac materials and display fluctuation-induced first-order transitions and multicritical behavior inherited from the tensor structure of the order parameter.

7. Operator Spectrum, Unitarity, and Numerical Bootstrap

The operator content and RG flow structure of the GNY model CFT are further elucidated using the conformal bootstrap (Erramilli et al., 2022, Mitchell et al., 18 Jun 2024):

• For all studied $N$ (e.g., $N=2,4,8$), only two relevant O(N) singlet scalar operators exist (both Δ < 3 in $d=3$); all subleading singlet scalars are irrelevant (Δ > 3).
• These results sharply constrain the RG flow structure and provide rigorous bounds for the universality class.
• Correction-to-scaling exponents $\omega_\pm$ are obtained both from high-loop field theory (Manashov et al., 2017, Gracey et al., 30 Jul 2025) and from bootstrap, with full agreement to order $1/N^2$ in $d$ dimensions.
• In non-integer dimensions, unitarity violation (negative-norm states) is possible due to evanescent operator mixing (Ji et al., 2018).

This rigorous mapping of the operator spectrum enhances our understanding of the universality and isolation of the fixed point in the space of 3d conformal field theories.

8. Nonequilibrium and Long-Range Extensions

GNY-type models have also been analyzed in nonequilibrium quenches, with rich prethermal dynamics controlled by distinct fixed points:

• Shallow quenches: universal non-equilibrium scaling controlled by the equilibrium Ising GNY fixed point.
• Deep quenches: non-thermal fixed point with gapless fermions (the "dynamical chiral Ising fixed point"), characterized by a finite Yukawa coupling and distinct RG scaling (Jian et al., 2019).

Long-range Gross–Neveu–Yukawa models (with nonlocal kinetic term) yield critical regimes in the UV or IR depending on the range parameter, providing a further laboratory for duality and conformal data equivalence studies (Chai et al., 2021).

9. Outlook and Theoretical Significance

The Gross-Neveu-Yukawa framework unifies a wide variety of continuous quantum phase transitions in interacting Dirac materials and relativistic fermion systems. The convergence of high-order perturbative RG, 1/N-expansion, nonperturbative functional RG, quantum Monte Carlo, and conformal bootstrap strongly corroborates the universality and operator structure of these fixed points. Recent multi-loop computations (now to five-loop (Gracey et al., 30 Jul 2025)) are approaching the precision necessary to settle longstanding discrepancies between analytic and numerical approaches, especially for low $N$ and physically relevant systems such as graphene.

Ongoing challenges include the extension of precision calculations to tensorial and matrix GNY models, further nonperturbative input for small $N_f$, full classification of boundary universality classes, and the investigation of GNY CFTs in non-integer dimensions, where subtle unitarity issues arise.

Overall, the GNY model remains a cornerstone in the paper of fermionic criticality, emergent symmetry, and universal phenomena in quantum many-body and field-theoretic systems.