---
title: Gross-Neveu-Yukawa Models Overview
url: https://www.emergentmind.com/topics/gross-neveu-yukawa-gny-models
type: topic
---

# Gross-Neveu-Yukawa Models Overview

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.) [2304.07654, 2512.11963].

The precise universality class is determined by the representation and number $M$ of the bosonic fields:
- **Chiral Ising ($M=1$):** Real scalar, $Z_2$ symmetry—semimetal$\rightarrow$CDW, surface transitions in topological insulators [1709.05057, 1707.05275, 2210.02492].
- **Chiral XY ($M=2$):** $O(2)$ vector, e.g., Kekulé VBS, SC transitions [2512.11963].
- **Chiral Heisenberg ($M=3$):** $O(3)$ vector, AFM/Néel transitions in graphene [2304.07654].
- **SO(2N)$\rightarrow$SO(N)$\times$SO(N):** Tensor order breaking, unifies mass-gap orders in generalized Dirac systems [2406.01681].

Global symmetry is at least $O(M)$ 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 ($v_f=v_b$) [2304.07654, 1910.07430].

## 2. Renormalization Group Structure and Critical Exponents

Wilson-Fisher $\epsilon$-expansion about $d=4$ provides controlled access to the RG flows and critical exponents. The one-loop $\beta$-functions for the dimensionless couplings $y = g^2/(8\pi^2)$ and $\lambda/(8\pi^2)$ are [1709.05057, 2304.07654, 2507.22594]:
\[
\begin{align*}
\beta_y &= -\epsilon y + (3-M+2N)y^2 + \cdots, \\
\beta_\lambda &= -\epsilon\lambda + (36+4M)\lambda^2 + 4N y \lambda - N y^2 + \cdots.
\end{align*}
\]
At the IR-stable fixed point, the leading (one-loop) values are
\[
y_* = \frac{\epsilon}{3+2N-M} + \mathcal{O}(\epsilon^2).
\]
Critical exponents at $d=4-\epsilon$ are, e.g., for chiral Ising ($M=0$) with $N$ fermion flavors [1709.05057, 2507.22594]:
\[
\begin{align*}
\eta_\psi &= y_*/2 + \mathcal{O}(y_*^2), \\
\eta_\phi &= 2N y_* + \mathcal{O}(y_*^2), \\
1/\nu &= 2 - \eta_\phi + \eta_{\phi^2}.
\end{align*}
\]
High-precision results up to five loops are available for all universal exponents, including the correction-to-scaling exponent $\omega$ [2507.22594].

Borel/Padé resummations of the $\epsilon$-series yield three-dimensional estimates in strong agreement with conformal bootstrap and QMC, e.g., for $N=2$ (graphene CDW transition) $\eta_\psi\approx0.04238(11),\;\eta_\phi\approx0.7329(27),\;1/\nu\approx0.998(12)$ [2507.22594, 2210.02492, 2304.00034].

## 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 $(x>0)$, for armchair-terminated honeycomb lattices, Dirichlet-type conditions on $\psi$ and either Dirichlet or Neumann on $\phi$ correspond to these three classes; each yields distinct RG flows and surface scaling dimensions [2503.13247, 2606.07510].

Surface critical exponents such as the boundary fermion or scalar scaling dimensions $\Delta_{\hat\psi},\;\Delta_{\hat\phi}$ are obtained via additional renormalization of boundary operators; at one loop,
\[
\Delta_{\hat\psi}^\text{(Dirichlet)} = \frac{d-1}{2} + \eta_{\hat\psi},\quad
\Delta_{\hat\phi}^\text{(Neumann)} = 1 + \eta_{\hat\phi}.
\]
Higher-order corrections and boundary central charges have been computed in both the $1/N$ and $\epsilon$ expansions, with full consistency between approaches in the large-$N$ limit [2606.07510].

### 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 ($z>1$) emerges generically in the disordered regime. Long-range correlated disorder leads to Weinrib-Halperin superuniversality, $\nu=2/\alpha$ [2008.13663].

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 $d=4-\epsilon$, the instanton action scales as $S_\text{inst}^{\rm IR} = \frac{2N}{3}\epsilon$, strongly suppressing non-perturbative effects as $N$ increases [2508.12080].

Large-$N$ correspondence between saddle-point computations and Hubbard-Stratonovich (auxiliary field) reformulations hold for both instanton actions and subleading corrections. Mapping to $S^d$ 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 $N$:
- **Chiral Ising, $N=1/4$:** Emergent $\mathcal{N}=1$ SUSY—one real scalar and one Majorana fermion. Ward identities hold: $\eta_\psi = \eta_\phi$, $1/\nu = (d-\eta)/2$ [1709.05057, 2604.10434].
- **Chiral XY, $N=1/2$:** Emergent $\mathcal{N}=2$ SUSY; $\eta_\psi = \eta_\phi$ and $\nu^{-1} = 2-\omega$ [1709.05057].
  
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 [2604.10434].

Further, systematic classification demonstrates all fixed-point types for $O(M)$-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 [2512.11963, 2406.01681].

## 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 $L\sim 36$ for the $N=8$ chiral Ising GNY class, with results in precise agreement with conformal bootstrap and $\epsilon$-expansion [2304.00034, 1910.07430].

The conformal bootstrap, leveraging mixed fermion-boson correlator spaces and SDP technology, has isolated disjoint "islands" of allowed scaling dimensions (“GNY archipelago”) for various $N$, tightly bounding operator scaling dimensions and OPE coefficients. For $N=2$, estimates are $\Delta_\psi \approx 1.06861(12)$, $\Delta_\sigma \approx 0.6500(12)$, $\Delta_\epsilon \approx 1.725(7)$ [2210.02492]. These match Borel-resummed high-loop RG and large-$N$ 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.) [2304.07654, 2210.02492].

## 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 [2304.07654, 1709.05057].
- **Surface transitions in topological insulators/superconductors:** Where time-reversal or spin-rotational symmetry is broken at the surface [2210.02492].
- **Designer systems via optical lattices and artificial Dirac materials:** Allowing clean tuning of coupling strengths and realization of armchair boundaries [2503.13247].

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 [2008.13663].

## 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 [2507.22594, 2210.02492].
- **Emergent symmetry and duality:** E.g., the deconfined Néel–VBS transition is conjectured to be dual to QED$_3$-GNY at $N=2$, with Aslamazov-Larkin diagrams crucial for matching critical exponents [1812.02720].
- **Disorder, multicriticality, and limit cycles:** Multicritical points, fixed-point annihilation, and Hopf bifurcations encode rich physics, including discrete scale invariance [2008.13663].
- **Thermodynamics and CFT at finite-T:** Interpolations between weak/strong coupling, entropy deficits, and trace anomaly behaviors, including the notable $s/s_\text{free}=31/35$ when $N_f=1/4$, paralleling SUSY WZ models [2007.03784].
- **Computational advances:** Exploiting SUSY Ward identities, advanced QMC, and symbolic packages to optimize high-order loop calculations and aid future precision studies [2604.10434, 1910.07430].

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 [2304.07654, 1709.05057, 2210.02492, 2606.07510, 2507.22594].

Source: https://www.emergentmind.com/topics/gross-neveu-yukawa-gny-models