Conformal Gross-Neveu-Yukawa Model Overview

• The conformal GNY model is a critical conformal field theory defined by tuning a Yukawa-coupled fermion–scalar system to quantum criticality.
• It employs multiple formulations, including the three-dimensional O(N) and 4-ε approaches, with analyses via bootstrap, 1/N expansion, and perturbative renormalization.
• High-precision studies using Monte Carlo and exact diagonalization reveal a detailed operator spectrum and establish the irrelevance of additional singlet deformations.

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 (Mitchell et al., 2024, Goykhman et al., 2020, Mihaila et al., 2017).

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 $\psi_i$, $N$0, two-component Majorana fermions in the fundamental of $N$1. The Yukawa term $N$2 contracts spinor indices with the two-dimensional charge-conjugation matrix, and the global symmetry acts only on the fermion index $N$3 (Mitchell et al., 2024).

A closely related $N$4-dimensional form, widely used in $N$5 and $N$6 analyses, is

$N$7

with a real scalar $N$8 and Dirac fermions $N$9. In the large-$O(N)$0 Gross–Neveu model one may instead start from

$O(N)$1

and introduce an auxiliary field $O(N)$2 through the Hubbard–Stratonovich replacement

$O(N)$3

At criticality in $O(N)$4, $O(N)$5 acquires a nontrivial two-point function induced by the fermion loop, and the GN and GNY fixed points coincide (Goykhman et al., 2020).

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)$6 symmetry, whereas many $O(N)$7-expansion papers use four-component Dirac fermions and write $O(N)$8 or $O(N)$9 for the fermion multiplicity (Erramilli et al., 2022, Mihaila et al., 2017).

2. Interacting fixed point and renormalization-group structure

In the three-dimensional $d=4-\epsilon$0 theory, tuning to $d=4-\epsilon$1 produces an interacting conformal fixed point for all finite $d=4-\epsilon$2. The leading scalar singlets are the parity-odd $d=4-\epsilon$3 and the parity-even $d=4-\epsilon$4, both with scaling dimension below $d=4-\epsilon$5, followed by $d=4-\epsilon$6 and $d=4-\epsilon$7 (Mitchell et al., 2024).

Near four dimensions, the model is perturbatively renormalizable and admits a nontrivial infrared fixed point. In the $d=4-\epsilon$8 scheme, one convenient two-loop presentation uses renormalized couplings $d=4-\epsilon$9 and $N$0 with

$N$1

$N$2

The simultaneous zeros define the GNY fixed point, with

$N$3

at leading order (Fei et al., 2016).

Higher-order perturbation theory substantially refines this picture. Three-loop calculations in $N$4 dimensions produced Padé estimates in $N$5 for several physically relevant flavor numbers; for example, in the convention of four-component Dirac fermions, the $N$6 estimates for $N$7 are $N$8, $N$9, and $N_f$0 (Mihaila et al., 2017). Five-loop renormalization extended the anomalous dimensions, $N_f$1-functions, and scalar mass operator to $N_f$2, and for $N_f$3 interpolating-polynomial estimates at $N_f$4 gave $N_f$5, $N_f$6, and $N_f$7 (Gracey et al., 30 Jul 2025).

A structurally important point is that the number of relevant singlet deformations equals the codimension of the critical manifold. Establishing the irrelevance of $N_f$8 and $N_f$9 therefore fixes the singlet-sector RG organization nonperturbatively (Mitchell et al., 2024).

3. Operator spectrum and conformal data

The operator content is commonly organized by parity, spin, and global-symmetry representation. In the three-dimensional $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,$0 model, $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,$1 are parity-odd singlets, while $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,$2 are parity-even singlets. Large-$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,$3 and $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,$4 analyses already indicate $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,$5, but the ordering at small $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,$6 requires nonperturbative control (Mitchell et al., 2024).

In the $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,$7 expansion for $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,$8, the basic low-lying dimensions are conventionally written as

$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,$9

with

$\phi$0

$\phi$1

These relations determine $\phi$2, $\phi$3, and $\phi$4 to next-to-leading order in $\phi$5 (Goykhman et al., 2020).

The same $\phi$6 program computes OPE data. Using conformal-triangle integrals, one obtains the normalized coefficients $\phi$7, $\phi$8, and $\phi$9 through next-to-leading order. In this framework, the dimensions $O(N)$0 together with the $O(N)$1 determine the CFT three-point functions and, through the bootstrap equations, constrain higher-point correlators (Goykhman et al., 2020).

This spectrum-based viewpoint is also central in three-dimensional precision studies. Mixed-correlator bootstrap analyses identify isolated islands for the leading dimensions $O(N)$2, while later work uses those islands as input to isolate the next singlet scalars $O(N)$3 and $O(N)$4 (Erramilli et al., 2022, Mitchell et al., 2024).

4. Mixed-correlator bootstrap and rigorous bounds in $O(N)$5

The three-dimensional numerical bootstrap for parity-preserving $O(N)$6-invariant GNY CFTs simultaneously studies

$O(N)$7

together with mixed correlators such as $O(N)$8, $O(N)$9, and $\psi_i$0. Crossing is cast in semidefinite-program form,

$\psi_i$1

and, after separating the identity and imposing positivity, as

$\psi_i$2

To bound $\psi_i$3 and $\psi_i$4, one imposes additional gaps $\psi_i$5 and $\psi_i$6, scans over the previously allowed $\psi_i$7 archipelago, and uses a “tiptop” search together with an explicit OPE-scan (Mitchell et al., 2024).

Earlier mixed-correlator work had already isolated small islands for $\psi_i$8. At $\psi_i$9 with $N$00, the rigorous estimates are

$N$01

$N$02

$N$03

For $N$04, without imposing supersymmetry, the island lies along $N$05 and has tip $N$06 (Erramilli et al., 2022).

The 2024 bootstrap study then proved that the next singlet scalars are irrelevant for $N$07 (Mitchell et al., 2024):

$N$08	lower bound on $N$09	lower bound on $N$10
2	$N$11	$N$12
4	$N$13	$N$14
8	$N$15	$N$16

These inequalities imply that there are only two relevant singlet scalars, $N$17 and $N$18, in the singlet sector at those values of $N$19. They also remove an assumption built into the earlier islands: the irrelevance of $N$20 and $N$21 is no longer an external bootstrap input but a rigorously established consequence. In particular, perturbations of $N$22-type or $N$23-type flow back to the same critical fixed point, so no additional multicritical fixed points appear in the singlet sector for $N$24 (Mitchell et al., 2024).

5. Supersymmetric, long-range, and classified extensions

The GNY family has distinguished supersymmetric limits. In the $N$25 Majorana case, formal continuation yields the supersymmetric $N$26 model, with couplings satisfying $N$27 through $N$28 and dimensions obeying $N$29 to the same order. A one-sided Padé extrapolation gives $N$30 in $N$31, in agreement with the $N$32 $N$33 minimal SCFT picture (Fei et al., 2016). Independent bootstrap evidence appears in the $N$34 archipelago, where the spectrum aligns along the supersymmetric relation $N$35 without imposing SUSY as an assumption (Erramilli et al., 2022).

A two-dimensional supersymmetric GNY theory has also been solved with Lightcone Conformal Truncation. There the model depends on a single dimensionless coupling $N$36, has a critical point at $N$37, and flows in the IR to the Tricritical Ising Model. A fit at $N$38 gives $N$39, consistent with the TIM value $N$40, while beyond the critical point the gap remains nearly zero, in agreement with a massless Goldstino on the SUSY-broken side (Fitzpatrick et al., 2019).

Long-range generalizations replace the local fermion kinetic term by a nonlocal one controlled by an exponent $N$41. 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,

$N$42

and

$N$43

coinciding term by term with the long-range GN data under $N$44 and $N$45 (Chai et al., 2021).

A broader 2025 classification of $N$46 GNY-like models with $N$47 real scalars and $N$48 symmetry found that nontrivial unitary fixed points with unbroken $N$49 occur only for $N$50. The resulting six universality classes are: for $N$51, standard $N$52 GNY, chiral Ising, and quarter-GNY; for $N$53, chiral XY; and for $N$54, chiral Heisenberg and the new orthogonal Heisenberg CFT with $N$55 symmetry (Mitchell et al., 12 Dec 2025).

6. Gauged and boundary extensions

A major gauged relative is the chiral QED$N$56–GNY theory, a $N$57-dimensional $N$58 gauge theory with $N$59 four-component Dirac fermions coupled to a scalar order parameter. In $N$60, the one-loop $N$61-functions are

$N$62

$N$63

There is a unique infrared-stable fixed point with all couplings nonzero when $N$64 and $N$65. For the chiral-Ising case $N$66, Padé $N$67 at $N$68 gives $N$69 and $N$70 for $N$71, and the proposed duality to the noncompact $N$72 deconfined critical point would require $N$73 (Zhou et al., 2020).

The small-$N$74 QED$N$75–GNY problem also illustrates a technical controversy. In strict $N$76 with two-component spinors, Aslamazov–Larkin diagrams contribute because

$N$77

These diagrams were missed in earlier $N$78 and large-$N$79 treatments with four-component fermions. Including them changes $N$80, $N$81, and the singlet bilinear dimension, while leaving the adjoint bilinear unchanged at $N$82. For $N$83, Padé and Borel–Padé resumations give $N$84 or $N$85, and $N$86 or $N$87, in reasonable agreement with numerical studies of the Néel–VBS transition (Boyack et al., 2018).

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: $N$88, corresponding to Neumann boundary conditions on the scalar; $N$89, corresponding to Dirichlet; and $N$90, in which the bulk scalar acquires a classical expectation value. The flows $N$91 and $N$92 are driven by relevant boundary deformations $N$93 and $N$94, and the AdS free energies satisfy the ordering expected from the boundary $N$95-theorem (Giombi et al., 2021).

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$96 chiral-Ising GNY critical point, the continuum action was taken as

$N$97

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 $N$98, $N$99, $O(N)$00, $O(N)$01, and $O(N)$02 on systems up to $O(N)$03. The Matsubara conductivity obeyed

$O(N)$04

and the designer model recovered the free-fermion value $O(N)$05 to $O(N)$06 accuracy (Liu et al., 2019).

Exact diagonalization on $O(N)$07 provides a complementary route to the operator spectrum. For a single Dirac cone, the sphere Dirac operator is

$O(N)$08

with eigenspinors given by spinor spherical harmonics. Many-body energies on $O(N)$09 are converted to scaling dimensions through

$O(N)$10

with the nonuniversal velocity $O(N)$11 fixed by $O(N)$12. In the $O(N)$13 ($O(N)$14) case, exact diagonalization at $O(N)$15 found $O(N)$16, $O(N)$17, $O(N)$18, $O(N)$19, $O(N)$20, $O(N)$21, and $O(N)$22, with the stress tensor fixed exactly at $O(N)$23. The same spectrum revealed two previously uncharacterized higher primaries, a nonconserved spin-1 current $O(N)$24 with $O(N)$25 and a composite fermion $O(N)$26 with $O(N)$27 (Gao et al., 21 Apr 2025).

Taken together, bootstrap islands, high-order $O(N)$28-expansions, $O(N)$29 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 $O(N)$30 $O(N)$31 singlet sector is that, for $O(N)$32, the only relevant singlet scalars are $O(N)$33 and $O(N)$34, while the next parity-odd and parity-even singlets are rigorously irrelevant (Mitchell et al., 2024).