---
title: Conformal Gross-Neveu-Yukawa Model Overview
url: https://www.emergentmind.com/topics/conformal-gross-neveu-yukawa-model
type: topic
---

# Conformal Gross-Neveu-Yukawa Model Overview

The conformal Gross–Neveu–Yukawa (GNY) model is the interacting conformal field theory obtained by tuning a Yukawa-coupled fermion–scalar system to criticality. In the three-dimensional \(O(N)\) formulation emphasized by mixed-correlator bootstrap studies, the theory contains a real scalar singlet and \(N\) Majorana fermions transforming in the fundamental of \(O(N)\); in \(d=4-\epsilon\) formulations it is often written with a real scalar and \(N\) or \(N_f\) Dirac fermions. At criticality it lies in the same universality class as the Gross–Neveu model after a Hubbard–Stratonovich transform, and it provides a conformal description of several fermionic quantum phase transitions together with supersymmetric, long-range, gauged, and boundary generalizations [2406.12974][2011.07768][1703.08801].

## 1. Field content, symmetries, and equivalent formulations

A standard three-dimensional Euclidean presentation is
\[
L=\tfrac12(\partial \phi)^2+\tfrac12\,\psi_i\,\slashed\partial\,\psi_i+\tfrac12\,m^2\phi^2+\tfrac14\,\lambda\,\phi^4+\tfrac12\,g\,\phi\,\psi_i\psi_i,
\]
with \(\phi\) a real scalar \(O(N)\) singlet and \(\psi_i\), \(i=1,\dots,N\), two-component Majorana fermions in the fundamental of \(O(N)\). The Yukawa term \(\phi\,\psi_i\psi_i\) contracts spinor indices with the two-dimensional charge-conjugation matrix, and the global symmetry acts only on the fermion index \(i\) [2406.12974].

A closely related \(d\)-dimensional form, widely used in \(4-\epsilon\) and \(1/N\) analyses, is
\[
S_{\mathrm{GNY}}=\int d^d x\left[\bar\psi^i\gamma^\mu\partial_\mu\psi^i+\tfrac12(\partial_\mu s)^2+g\,s\,\bar\psi^i\psi^i+\frac{\lambda}{4!}s^4\right],
\]
with a real scalar \(s\) and Dirac fermions \(\psi^i\). In the large-\(N\) Gross–Neveu model one may instead start from
\[
S_{\mathrm{GN}}=\int d^d x\left[\bar\psi^i\gamma^\mu\partial_\mu\psi^i+\frac{G}{N}(\bar\psi^i\psi^i)^2\right]
\]
and introduce an auxiliary field \(s\) through the Hubbard–Stratonovich replacement
\[
(\bar\psi\psi)^2\rightarrow-\frac{1}{4G}s^2+\frac{1}{\sqrt N}s\,\bar\psi\psi.
\]
At criticality in \(2<d<4\), \(s\) acquires a nontrivial two-point function induced by the fermion loop, and the GN and GNY fixed points coincide [2011.07768].

The literature summarized here uses more than one flavor convention. Three-dimensional bootstrap work typically parameterizes the theory by Majorana fermions and an \(O(N)\) symmetry, whereas many \(\epsilon\)-expansion papers use four-component Dirac fermions and write \(N\) or \(N_f\) for the fermion multiplicity [2210.02492][1703.08801].

## 2. Interacting fixed point and renormalization-group structure

In the three-dimensional \(O(N)\) theory, tuning to \(m^2=m_c^2(g,\lambda)\) produces an interacting conformal fixed point for all finite \(N\). The leading scalar singlets are the parity-odd \(\sigma\sim\phi\) and the parity-even \(\epsilon\sim\phi^2\), both with scaling dimension below \(3\), followed by \(\sigma'\sim\phi^3\) and \(\epsilon'\sim\phi\partial^2\phi\) [2406.12974].

Near four dimensions, the model is perturbatively renormalizable and admits a nontrivial infrared fixed point. In the \(\overline{\mathrm{MS}}\) scheme, one convenient two-loop presentation uses renormalized couplings \(y\) and \(\lambda\) with
\[
\beta_y=-\tfrac12\,\epsilon\,y+\frac{N+6}{2(4\pi)^2}y^3+\frac{1}{(4\pi)^4}\left[-\tfrac34(4N+3)y^5-2y^3\lambda+\tfrac1{12}y\lambda^2\right]+\cdots,
\]
\[
\beta_\lambda=-\epsilon\,\lambda+\frac{1}{(4\pi)^2}\left[3\lambda^2+2Ny^2\lambda-12Ny^4\right]+\frac{1}{(4\pi)^4}\left[-\tfrac{17}{3}\lambda^3-3Ny^2\lambda^2+7Ny^4\lambda+96Ny^6\right]+\cdots.
\]
The simultaneous zeros define the GNY fixed point, with
\[
y^{*2}(4\pi)^2=\frac{\epsilon}{N+6}+O(\epsilon^2)
\]
at leading order [1607.05316].

Higher-order perturbation theory substantially refines this picture. Three-loop calculations in \(4-\epsilon\) dimensions produced Padé estimates in \(d=3\) for several physically relevant flavor numbers; for example, in the convention of four-component Dirac fermions, the \(P_{[2/1]}\) estimates for \(N=2\) are \(\nu^{-1}\simeq1.048\), \(\eta_\phi\simeq0.672\), and \(\eta_\psi\simeq0.0740\) [1703.08801]. Five-loop renormalization extended the anomalous dimensions, \(\beta\)-functions, and scalar mass operator to \(\mathcal O(\epsilon^5)\), and for \(N=2\) interpolating-polynomial estimates at \(d=3\) gave \(\eta_\psi\simeq0.0414\), \(\eta_\phi\simeq0.7345\), and \(1/\nu\simeq1.0213\) [2507.22594].

A structurally important point is that the number of relevant singlet deformations equals the codimension of the critical manifold. Establishing the irrelevance of \(\sigma'\) and \(\epsilon'\) therefore fixes the singlet-sector RG organization nonperturbatively [2406.12974].

## 3. Operator spectrum and conformal data

The operator content is commonly organized by parity, spin, and global-symmetry representation. In the three-dimensional \(O(N)\) model, \(\sigma,\sigma',\sigma''\) are parity-odd singlets, while \(\epsilon,\epsilon',\epsilon''\) are parity-even singlets. Large-\(N\) and \(4-\epsilon\) analyses already indicate \(\Delta_{\sigma'},\Delta_{\epsilon'}>3\), but the ordering at small \(N\) requires nonperturbative control [2406.12974].

In the \(1/N\) expansion for \(2<d<4\), the basic low-lying dimensions are conventionally written as
\[
\Delta_\psi=\frac{d-1}{2}+\gamma_\psi,\qquad \Delta_s=1+\gamma_s,\qquad \Delta_{s^2}=2+\gamma_{s^2},
\]
with
\[
\gamma_\psi=-\frac1N\frac{2^{d-1}\sin(\pi d/2)\Gamma((d-1)/2)}{\pi^{3/2}d\,\Gamma(d/2-1)}+O(1/N^2),
\]
\[
\gamma_s=-4\frac{d-1}{d-2}\gamma_\psi+O(1/N^2),\qquad
\gamma_{s^2}=(2-d)\gamma_s+O(1/N^2).
\]
These relations determine \(\Delta_\psi\), \(\Delta_s\), and \(\Delta_{s^2}\) to next-to-leading order in \(1/N\) [2011.07768].

The same \(1/N\) program computes OPE data. Using conformal-triangle integrals, one obtains the normalized coefficients \(\widehat C_{\psi\psi s}\), \(\widehat C_{s^2ss}\), and \(\widehat C_{s^2\psi\psi}\) through next-to-leading order. In this framework, the dimensions \(\Delta_i\) together with the \(C_{ijk}\) determine the CFT three-point functions and, through the bootstrap equations, constrain higher-point correlators [2011.07768].

This spectrum-based viewpoint is also central in three-dimensional precision studies. Mixed-correlator bootstrap analyses identify isolated islands for the leading dimensions \((\Delta_\psi,\Delta_\sigma,\Delta_\epsilon)\), while later work uses those islands as input to isolate the next singlet scalars \(\sigma'\) and \(\epsilon'\) [2210.02492][2406.12974].

## 4. Mixed-correlator bootstrap and rigorous bounds in \(d=3\)

The three-dimensional numerical bootstrap for parity-preserving \(O(N)\)-invariant GNY CFTs simultaneously studies
\[
\langle \psi\psi\psi\psi\rangle,\quad
\langle \sigma\sigma\sigma\sigma\rangle,\quad
\langle \epsilon\epsilon\epsilon\epsilon\rangle,
\]
together with mixed correlators such as \(\langle\psi\psi\sigma\sigma\rangle\), \(\langle\psi\psi\epsilon\epsilon\rangle\), and \(\langle\sigma\sigma\epsilon\epsilon\rangle\). Crossing is cast in semidefinite-program form,
\[
\sum_O \mathbf{\lambda}_O^T\,\mathcal F_O\,\mathbf{\lambda}_O=0,
\]
and, after separating the identity and imposing positivity, as
\[
\alpha[\mathbb 1]+\sum_\rho\sum_{\Delta\ge \Delta_{\min}(\rho)}
\mathrm{Tr}\!\left[P^\rho_\Delta\,\alpha[F_{\Delta,\rho}]\right]=0.
\]
To bound \(\sigma'\) and \(\epsilon'\), one imposes additional gaps \(\Delta_{\sigma''}>4.5\) and \(\Delta_{\epsilon''}>4\), scans over the previously allowed \((\Delta_\psi,\Delta_\sigma,\Delta_\epsilon)\) archipelago, and uses a “tiptop” search together with an explicit OPE-scan [2406.12974].

Earlier mixed-correlator work had already isolated small islands for \(N=1,2,4,8\). At \(n_{\max}=18\) with \(\Delta_{\sigma'}>3\), the rigorous estimates are
\[
N=2:\ (\Delta_\psi,\Delta_\sigma,\Delta_\epsilon)=(1.06861(12),\,0.6500(12),\,1.725(7)),
\]
\[
N=4:\ (\Delta_\psi,\Delta_\sigma,\Delta_\epsilon)=(1.04356(16),\,0.7578(15),\,1.899(10)),
\]
\[
N=8:\ (\Delta_\psi,\Delta_\sigma,\Delta_\epsilon)=(1.02119(5),\,0.8665(13),\,2.002(12)).
\]
For \(N=1\), without imposing supersymmetry, the island lies along \(\Delta_\epsilon=\Delta_\psi+1/2=\Delta_\sigma+1\) and has tip \(\Delta_\sigma=0.58444(8)\) [2210.02492].

The 2024 bootstrap study then proved that the next singlet scalars are irrelevant for \(N=2,4,8\) [2406.12974]:

| \(N\) | lower bound on \(\Delta_{\sigma'}\) | lower bound on \(\Delta_{\epsilon'}\) |
|---|---:|---:|
| 2 | \(>3.071\) | \(>3.169\) |
| 4 | \(>3.241\) | \(>3.137\) |
| 8 | \(>3.189\) | \(>3.167\) |

These inequalities imply that there are only two relevant singlet scalars, \(\sigma\) and \(\epsilon\), in the singlet sector at those values of \(N\). They also remove an assumption built into the earlier islands: the irrelevance of \(\sigma'\) and \(\epsilon'\) is no longer an external bootstrap input but a rigorously established consequence. In particular, perturbations of \(\phi^3\)-type or \(\phi\partial^2\phi\)-type flow back to the same critical fixed point, so no additional multicritical fixed points appear in the singlet sector for \(N=2,4,8\) [2406.12974].

## 5. Supersymmetric, long-range, and classified extensions

The GNY family has distinguished supersymmetric limits. In the \(N=1\) Majorana case, formal continuation yields the supersymmetric \(W\sim\Sigma^3\) model, with couplings satisfying \(g_2=3g_1^2\) through \(O(\epsilon^2)\) and dimensions obeying \(\Delta_\sigma=\Delta_\psi-\tfrac12\) to the same order. A one-sided Padé extrapolation gives \(\Delta_\sigma\approx0.59\) in \(d=3\), in agreement with the \(3d\) \(\mathcal N=1\) minimal SCFT picture [1607.05316]. Independent bootstrap evidence appears in the \(N=1\) archipelago, where the spectrum aligns along the supersymmetric relation \(\Delta_\epsilon=\Delta_\psi+1/2=\Delta_\sigma+1\) without imposing SUSY as an assumption [2210.02492].

A two-dimensional supersymmetric GNY theory has also been solved with Lightcone Conformal Truncation. There the model depends on a single dimensionless coupling \(\bar g=g/m\), has a critical point at \(\bar g_*\approx1.5\), and flows in the IR to the Tricritical Ising Model. A fit at \(\Delta_{\max}=28\) gives \(\nu=1.24\pm0.05\), consistent with the TIM value \(5/4\), while beyond the critical point the gap remains nearly zero, in agreement with a massless Goldstino on the SUSY-broken side [1911.10220].

Long-range generalizations replace the local fermion kinetic term by a nonlocal one controlled by an exponent \(s\). In that setting, the long-range GN model flows in the UV to a nontrivial critical CFT while the long-range GNY model flows in the IR to the same CFT. The matching holds for scaling dimensions and cubic OPE coefficients: for example,
\[
\Delta_\theta=\frac{d-s+1}{2}\quad\text{exactly},\qquad
\Delta_\phi=s-1-\frac{4\epsilon_1}{N}+O(\epsilon^2,1/N^2),
\]
and
\[
C_{\phi\theta\theta}=-\sqrt{\frac{2\epsilon_1}{N}}\left(1-\frac2N+\cdots\right)+O(\epsilon^{3/2}),
\]
coinciding term by term with the long-range GN data under \(\psi\leftrightarrow\theta\) and \(\sigma\leftrightarrow\phi\) [2110.00020].

A broader 2025 classification of \((2+1)d\) GNY-like models with \(M\) real scalars and \(O(M)\) symmetry found that nontrivial unitary fixed points with unbroken \(O(M)\) occur only for \(M\le 3\). The resulting six universality classes are: for \(M=1\), standard \(O(4N_f)\) GNY, chiral Ising, and quarter-GNY; for \(M=2\), chiral XY; and for \(M=3\), chiral Heisenberg and the new orthogonal Heisenberg CFT with \(O(N_f)\times O(3)\) symmetry [2512.11963].

## 6. Gauged and boundary extensions

A major gauged relative is the chiral QED\(_3\)–GNY theory, a \(2+1\)-dimensional \(U(1)\) gauge theory with \(N_f\) four-component Dirac fermions coupled to a scalar order parameter. In \(d=4-\epsilon\), the one-loop \(\beta\)-functions are
\[
\beta_e=-\tfrac\epsilon2\,e+\frac{N_f}{12\pi^2}e^3,\qquad
\beta_g=-\tfrac\epsilon2\,g+\frac{2N_f-N_b+4}{16\pi^2}g^3-\frac{6}{16\pi^2}g\,e^2,
\]
\[
\beta_\lambda=-\epsilon\lambda+\frac{N_b+8}{3(4\pi)^2}\lambda^2+\frac{8N_f}{(4\pi)^2}g^2\lambda-\frac{48N_f}{(4\pi)^2}g^4.
\]
There is a unique infrared-stable fixed point with all couplings nonzero when \(2N_f-N_b+4>0\) and \(\lambda_*^+>0\). For the chiral-Ising case \(N_b=1\), Padé \([0/1]\) at \(\epsilon=1\) gives \(\eta_\phi\approx2.20\) and \(\nu^{-1}\approx0.677\) for \(N_f=1\), and the proposed duality to the noncompact \(\mathrm{CP}^1\) deconfined critical point would require \(\nu^{-1}>0.75\) [2004.04612].

The small-\(N\) QED\(_3\)–GNY problem also illustrates a technical controversy. In strict \(d=3\) with two-component spinors, Aslamazov–Larkin diagrams contribute because
\[
\mathrm{tr}\,\gamma_\mu\gamma_\nu\gamma_\rho=2i\,\epsilon_{\mu\nu\rho}.
\]
These diagrams were missed in earlier \(4-\epsilon\) and large-\(N\) treatments with four-component fermions. Including them changes \(\eta_\phi\), \(\nu^{-1}\), and the singlet bilinear dimension, while leaving the adjoint bilinear unchanged at \(\mathcal O(1/N)\). For \(N=2\), Padé and Borel–Padé resumations give \(\eta_\phi\approx0.17\) or \(0.30\), and \(\Delta_{\bar\psi T_A\psi}\approx1.42\) or \(1.51\), in reasonable agreement with numerical studies of the Néel–VBS transition [1812.02720].

Boundary criticality provides another extension. By a Weyl transformation, half-space Gross–Neveu criticality can be studied in Euclidean AdS. Near four dimensions, the Wilson–Fisher fixed point of the GNY model has three boundary conformal phases: \(B_2'\), corresponding to Neumann boundary conditions on the scalar; \(B_2\), corresponding to Dirichlet; and \(B_1\), in which the bulk scalar acquires a classical expectation value. The flows \(B_2'\to B_2\) and \(B_2\to B_1\) are driven by relevant boundary deformations \(\int_\partial (\hat s)^2\) and \(\int_\partial \hat s\), and the AdS free energies satisfy the ordering expected from the boundary \(F\)-theorem [2110.04268].

## 7. Numerical realizations and direct spectroscopy

Large-scale numerical simulation has made the conformal GNY model directly accessible. In a designer quantum Monte Carlo study of the \(N_f=8\) chiral-Ising GNY critical point, the continuum action was taken as
\[
S_E=\int d^3x\left[\tfrac12(\partial_\mu\phi)^2+\tfrac12 r\phi^2+\frac{u}{4!}\phi^4+\bar\psi_i\gamma_\mu\partial_\mu\psi_i+g\,\phi\,\bar\psi_i\psi_i\right].
\]
Model design enlarged the ultraviolet linear-dispersion region and matched bare boson and fermion velocities to suppress finite-size drift. Using projector QMC together with self-learning updates, the study found \(m_c^*=-0.062(3)\), \(1/\nu=1.0(1)\), \(\eta_\phi=0.59(2)\), \(\eta_\psi=0.05(2)\), and \(\omega=0.8(1)\) on systems up to \(16\times16\). The Matsubara conductivity obeyed
\[
\sigma(i\omega_n)/\sigma_Q=\sigma_\infty+b_1(T/\omega_n)^{3-1/\nu}+b_2(T/\omega_n)^3+\cdots,
\]
and the designer model recovered the free-fermion value \(\sigma_\infty=1/4\) to \(\lesssim1\%\) accuracy [1910.07430].

Exact diagonalization on \(\mathbb S^2\) provides a complementary route to the operator spectrum. For a single Dirac cone, the sphere Dirac operator is
\[
D=\frac1R\bigl(\boldsymbol{\sigma}\!\cdot\!\mathbf L+1\bigr),
\]
with eigenspinors given by spinor spherical harmonics. Many-body energies on \(\mathbb S^2\times\mathbb R\) are converted to scaling dimensions through
\[
\Delta_n=R\,\delta E_n/v,
\]
with the nonuniversal velocity \(v\) fixed by \(\Delta(T_{\mu\nu})=3\). In the \(N_f=1\) (\(N=2\)) case, exact diagonalization at \(j_{\max}=\tfrac52\) found \(\Delta_\sigma=0.65\), \(\Delta_{\partial_\mu\sigma}=1.67\), \(\Delta_\psi=1.05\), \(\Delta_{\partial\psi}=1.95\), \(\Delta_{\sigma_T}\sim2.51\), \(\Delta_\epsilon\sim2.18\), and \(\Delta_{J_\mu}\sim2.29\), with the stress tensor fixed exactly at \(3.00\). The same spectrum revealed two previously uncharacterized higher primaries, a nonconserved spin-1 current \(\tilde J_\mu\) with \(\Delta\approx2.4\) and a composite fermion \(\tilde\psi\sim\phi\psi\) with \(\Delta\approx2.2\) [2504.15338].

Taken together, bootstrap islands, high-order \(\epsilon\)-expansions, \(1/N\) CFT data, Monte Carlo, and spherical exact diagonalization now provide a tightly constrained picture of conformal GNY criticality. The most robust nonperturbative conclusion in the standard \(3d\) \(O(N)\) singlet sector is that, for \(N=2,4,8\), the only relevant singlet scalars are \(\sigma\) and \(\epsilon\), while the next parity-odd and parity-even singlets are rigorously irrelevant [2406.12974].

Source: https://www.emergentmind.com/topics/conformal-gross-neveu-yukawa-model