Gross-Neveu-Yukawa Models Overview

• Gross-Neveu-Yukawa models are defined as quantum field theories coupling Dirac or Majorana fermions with critical bosonic fields through Yukawa interactions, exemplifying quantum phase transitions in semimetals.
• They leverage renormalization group techniques and high-loop computations to yield precise critical exponents and scaling dimensions for universality classes such as chiral Ising, XY, and Heisenberg.
• Numerical methods and conformal bootstrap studies validate the models' predictions, establishing bridges between field theory, experimental condensed matter transitions, and emergent symmetry phenomena.

Gross-Neveu-Yukawa (GNY) models are interacting quantum field theories of Dirac or Majorana fermions coupled via a Yukawa interaction to critical bosonic order-parameter fields, typically equipped with quartic self-interactions. Canonical GNY models describe a range of universality classes for quantum phase transitions of Dirac and Weyl semimetals, such as the chiral Ising, XY, and Heisenberg types. The models are UV-complete for $d<4$ and provide a controlled framework for analytic, numerical, and conformal bootstrap studies of strongly correlated gapless fermionic criticality in 2+1 and 3+1 dimensions. Their rich operator content, RG structure, and nonperturbative solutions also illuminate broader aspects of emergent symmetry, supersymmetry, boundary criticality, and disorder-induced phenomena.

1. Model Definition, Symmetries, and Universality Classes

A typical GNY model in $d=4-\epsilon$ dimensions comprises $N$ four-component Dirac fermions $\psi_i$ (or $N$ two-component Majoranas), coupled to a real or vector bosonic field $\phi^a$ via a Yukawa term and with quartic bosonic self-interactions. The Euclidean action generically takes the form

$$S = \int d^d x \left[ \bar\psi_i \gamma^\mu \partial_\mu \psi_i + g\,\phi^a\,\bar\psi_i \mathcal{M}^a \psi_i + \frac{1}{2}(\partial_\mu \phi^a)^2 + \frac{1}{4!}\lambda_{abcd} \phi^a\phi^b\phi^c\phi^d \right]$$

where $\mathcal{M}^a$ encodes the symmetry channel of the coupling (scalar for chiral Ising, vector for chiral Heisenberg, complex for chiral XY, etc.) (Herbut, 2023, Mitchell et al., 12 Dec 2025).

The precise universality class is determined by the representation and number $M$ of the bosonic fields:

• Chiral Ising ($M=1$): Real scalar, $d=4-\epsilon$0 symmetry—semimetal$d=4-\epsilon$1CDW, surface transitions in topological insulators (Zerf et al., 2017, Gracey, 2017, Erramilli et al., 2022).
• Chiral XY ($d=4-\epsilon$2): $d=4-\epsilon$3 vector, e.g., Kekulé VBS, SC transitions (Mitchell et al., 12 Dec 2025).
• Chiral Heisenberg ($d=4-\epsilon$4): $d=4-\epsilon$5 vector, AFM/Néel transitions in graphene (Herbut, 2023).
• SO(2N)$d=4-\epsilon$6SO(N)$d=4-\epsilon$7SO(N): Tensor order breaking, unifies mass-gap orders in generalized Dirac systems (Han et al., 2024).

Global symmetry is at least $d=4-\epsilon$8 on the bosonic sector and may be further enlarged by the combined boson-fermion symmetry group; emergent Lorentz invariance is realized at the IR fixed point ($d=4-\epsilon$9) (Herbut, 2023, Liu et al., 2019).

2. Renormalization Group Structure and Critical Exponents

Wilson-Fisher $N$0-expansion about $N$1 provides controlled access to the RG flows and critical exponents. The one-loop $N$2-functions for the dimensionless couplings $N$3 and $N$4 are (Zerf et al., 2017, Herbut, 2023, Gracey et al., 30 Jul 2025): $N$5 At the IR-stable fixed point, the leading (one-loop) values are

$N$6

Critical exponents at $N$7 are, e.g., for chiral Ising ($N$8) with $N$9 fermion flavors (Zerf et al., 2017, Gracey et al., 30 Jul 2025): $\psi_i$0 High-precision results up to five loops are available for all universal exponents, including the correction-to-scaling exponent $\psi_i$1 (Gracey et al., 30 Jul 2025).

Borel/Padé resummations of the $\psi_i$2-series yield three-dimensional estimates in strong agreement with conformal bootstrap and QMC, e.g., for $\psi_i$3 (graphene CDW transition) $\psi_i$4 (Gracey et al., 30 Jul 2025, Erramilli et al., 2022, Wang et al., 2023).

3. Boundary, Disorder, and Nonperturbative Phenomena

Boundary Criticality

GNY models exhibit rich boundary universality classes (“ordinary,” “special,” “normal/extraordinary”) depending on the boundary conditions imposed on fermion and boson fields. On a half-space $\psi_i$5, for armchair-terminated honeycomb lattices, Dirichlet-type conditions on $\psi_i$6 and either Dirichlet or Neumann on $\psi_i$7 correspond to these three classes; each yields distinct RG flows and surface scaling dimensions (Jiang et al., 17 Mar 2025, Diatlyk et al., 5 Jun 2026).

Surface critical exponents such as the boundary fermion or scalar scaling dimensions $\psi_i$8 are obtained via additional renormalization of boundary operators; at one loop,

$\psi_i$9

Higher-order corrections and boundary central charges have been computed in both the $N$0 and $N$1 expansions, with full consistency between approaches in the large-$N$2 limit (Diatlyk et al., 5 Jun 2026).

Quenched Disorder

Random-mass disorder modifies the critical properties, giving rise to new universal finite-randomness fixed points with critical exponents that depend on the range and strength of disorder. The RG equations involve additional disorder couplings, and non-Lorentz-invariant dynamics ($N$3) emerges generically in the disordered regime. Long-range correlated disorder leads to Weinrib-Halperin superuniversality, $N$4 (Yerzhakov et al., 2020).

Bifurcations such as transcritical exchanges, fixed-point annihilation (“walking”/Miransky scaling), and Hopf bifurcation (emergent limit cycles and discrete scale invariance) arise as function of disorder exponent and fermion flavor number.

Instanton and Nonperturbative Sectors

Euclidean instantons—finite-action, localized solutions to the GNY field equations—can be explicitly constructed. At the IR Wilson-Fisher fixed point in $N$5, the instanton action scales as $N$6, strongly suppressing non-perturbative effects as $N$7 increases (Imaanpur et al., 16 Aug 2025).

Large-$N$8 correspondence between saddle-point computations and Hubbard-Stratonovich (auxiliary field) reformulations hold for both instanton actions and subleading corrections. Mapping to $N$9 via stereographic projection confirms conformal invariance of the resulting semiclassical configurations.

4. Emergent Supersymmetry and Generalized GNY Models

GNY models can realize emergent supersymmetry at special values of $\phi^a$0:

• Chiral Ising, $\phi^a$1: Emergent $\phi^a$2 SUSY—one real scalar and one Majorana fermion. Ward identities hold: $\phi^a$3, $\phi^a$4 (Zerf et al., 2017, Chakraborty et al., 12 Apr 2026).
• Chiral XY, $\phi^a$5: Emergent $\phi^a$6 SUSY; $\phi^a$7 and $\phi^a$8 (Zerf et al., 2017).

A general unified Lagrangian framework simultaneously containing GNY, Nambu–Jona-Lasinio–Yukawa (NJLY), and Wess-Zumino (WZ) models clarifies the underlying algebraic structures and demonstrates how SUSY Ward identities can be leveraged to eliminate complex loop integrals even in non-supersymmetric settings (Chakraborty et al., 12 Apr 2026).

Further, systematic classification demonstrates all fixed-point types for $\phi^a$9-symmetric multifield GNY-like models, including the chiral Ising, XY, Heisenberg, and the new "orthogonal Heisenberg" CFT, as well as generalized tensor fixed points relevant for SO(2N) flavor symmetry breaking (Mitchell et al., 12 Dec 2025, Han et al., 2024).

5. Numerical and Conformal Bootstrap Studies

Quantum Monte Carlo (QMC), especially using lattice actions carefully engineered to minimize finite-size and velocity-mismatch artifacts, allows accurate extraction of critical exponents. The elective-momentum ultra-size (EMUS) QMC method achieves reliable scaling for system sizes up to $$S = \int d^d x \left[ \bar\psi_i \gamma^\mu \partial_\mu \psi_i + g\,\phi^a\,\bar\psi_i \mathcal{M}^a \psi_i + \frac{1}{2}(\partial_\mu \phi^a)^2 + \frac{1}{4!}\lambda_{abcd} \phi^a\phi^b\phi^c\phi^d \right]$$0 for the $$S = \int d^d x \left[ \bar\psi_i \gamma^\mu \partial_\mu \psi_i + g\,\phi^a\,\bar\psi_i \mathcal{M}^a \psi_i + \frac{1}{2}(\partial_\mu \phi^a)^2 + \frac{1}{4!}\lambda_{abcd} \phi^a\phi^b\phi^c\phi^d \right]$$1 chiral Ising GNY class, with results in precise agreement with conformal bootstrap and $$S = \int d^d x \left[ \bar\psi_i \gamma^\mu \partial_\mu \psi_i + g\,\phi^a\,\bar\psi_i \mathcal{M}^a \psi_i + \frac{1}{2}(\partial_\mu \phi^a)^2 + \frac{1}{4!}\lambda_{abcd} \phi^a\phi^b\phi^c\phi^d \right]$$2-expansion (Wang et al., 2023, Liu et al., 2019).

The conformal bootstrap, leveraging mixed fermion-boson correlator spaces and SDP technology, has isolated disjoint "islands" of allowed scaling dimensions (“GNY archipelago”) for various $$S = \int d^d x \left[ \bar\psi_i \gamma^\mu \partial_\mu \psi_i + g\,\phi^a\,\bar\psi_i \mathcal{M}^a \psi_i + \frac{1}{2}(\partial_\mu \phi^a)^2 + \frac{1}{4!}\lambda_{abcd} \phi^a\phi^b\phi^c\phi^d \right]$$3, tightly bounding operator scaling dimensions and OPE coefficients. For $$S = \int d^d x \left[ \bar\psi_i \gamma^\mu \partial_\mu \psi_i + g\,\phi^a\,\bar\psi_i \mathcal{M}^a \psi_i + \frac{1}{2}(\partial_\mu \phi^a)^2 + \frac{1}{4!}\lambda_{abcd} \phi^a\phi^b\phi^c\phi^d \right]$$4, estimates are $$S = \int d^d x \left[ \bar\psi_i \gamma^\mu \partial_\mu \psi_i + g\,\phi^a\,\bar\psi_i \mathcal{M}^a \psi_i + \frac{1}{2}(\partial_\mu \phi^a)^2 + \frac{1}{4!}\lambda_{abcd} \phi^a\phi^b\phi^c\phi^d \right]$$5, $$S = \int d^d x \left[ \bar\psi_i \gamma^\mu \partial_\mu \psi_i + g\,\phi^a\,\bar\psi_i \mathcal{M}^a \psi_i + \frac{1}{2}(\partial_\mu \phi^a)^2 + \frac{1}{4!}\lambda_{abcd} \phi^a\phi^b\phi^c\phi^d \right]$$6, $$S = \int d^d x \left[ \bar\psi_i \gamma^\mu \partial_\mu \psi_i + g\,\phi^a\,\bar\psi_i \mathcal{M}^a \psi_i + \frac{1}{2}(\partial_\mu \phi^a)^2 + \frac{1}{4!}\lambda_{abcd} \phi^a\phi^b\phi^c\phi^d \right]$$7 (Erramilli et al., 2022). These match Borel-resummed high-loop RG and large-$$S = \int d^d x \left[ \bar\psi_i \gamma^\mu \partial_\mu \psi_i + g\,\phi^a\,\bar\psi_i \mathcal{M}^a \psi_i + \frac{1}{2}(\partial_\mu \phi^a)^2 + \frac{1}{4!}\lambda_{abcd} \phi^a\phi^b\phi^c\phi^d \right]$$8 expansions to better than 1%.

Such numerical/analytic synergy provides benchmark data for experimental and theoretical tests (e.g., CDW and Néel transitions in (spinful/spinless) graphene, surface transitions in topological insulators, etc.) (Herbut, 2023, Erramilli et al., 2022).

6. Physical Realizations and Experimental Probes

GNY quantum critical points arise at interaction-tuned transitions in Dirac and Weyl semimetals, e.g.:

• Graphene CDW and AFM transitions: Mott and magnetic quantum critical points in honeycomb and related lattices (Herbut, 2023, Zerf et al., 2017).
• Surface transitions in topological insulators/superconductors: Where time-reversal or spin-rotational symmetry is broken at the surface (Erramilli et al., 2022).
• Designer systems via optical lattices and artificial Dirac materials: Allowing clean tuning of coupling strengths and realization of armchair boundaries (Jiang et al., 17 Mar 2025).

Boundary and bulk critical exponents can be probed via tunneling spectroscopy, edge-state power-laws in STM, or scaling of the local density of states. Experimental tests of disorder-induced fixed points or log-periodic oscillations (“walking”) require controlled disorder or correlated impurity engineering (Yerzhakov et al., 2020).

7. Extensions and Open Directions

GNY theory is a platform for a broad set of research directions:

• Higher-order RG and conformal data: Ongoing five-loop computations and bootstrap studies continually refine exponents and operator spectra (Gracey et al., 30 Jul 2025, Erramilli et al., 2022).
• Emergent symmetry and duality: E.g., the deconfined Néel–VBS transition is conjectured to be dual to QED$$S = \int d^d x \left[ \bar\psi_i \gamma^\mu \partial_\mu \psi_i + g\,\phi^a\,\bar\psi_i \mathcal{M}^a \psi_i + \frac{1}{2}(\partial_\mu \phi^a)^2 + \frac{1}{4!}\lambda_{abcd} \phi^a\phi^b\phi^c\phi^d \right]$$9-GNY at $\mathcal{M}^a$0, with Aslamazov-Larkin diagrams crucial for matching critical exponents (Boyack et al., 2018).
• Disorder, multicriticality, and limit cycles: Multicritical points, fixed-point annihilation, and Hopf bifurcations encode rich physics, including discrete scale invariance (Yerzhakov et al., 2020).
• Thermodynamics and CFT at finite-T: Interpolations between weak/strong coupling, entropy deficits, and trace anomaly behaviors, including the notable $\mathcal{M}^a$1 when $\mathcal{M}^a$2, paralleling SUSY WZ models (Pinto, 2020).
• Computational advances: Exploiting SUSY Ward identities, advanced QMC, and symbolic packages to optimize high-order loop calculations and aid future precision studies (Chakraborty et al., 12 Apr 2026, Liu et al., 2019).

Gross-Neveu-Yukawa models thus serve as a universal theoretical laboratory for exploring strongly-correlated gapless fermionic matter, emergent symmetries, and quantum phase transitions, with continuing relevance for both condensed matter and quantum field theory (Herbut, 2023, Zerf et al., 2017, Erramilli et al., 2022, Diatlyk et al., 5 Jun 2026, Gracey et al., 30 Jul 2025).