---
title: Boundary Gross-Neveu-Yukawa Fixed Point
url: https://www.emergentmind.com/topics/boundary-gross-neveu-yukawa-fixed-point
type: topic
---

# Boundary Gross-Neveu-Yukawa Fixed Point

Searching arXiv for recent papers on boundary Gross–Neveu–Yukawa fixed points and related BCFT work.
Boundary Gross–Neveu–Yukawa (GNY) fixed points are conformal boundary conditions of the GNY universality class, defined for a Yukawa theory on a half-space or, equivalently, on Euclidean anti de Sitter space through a Weyl map. Unlike the bulk GNY conformal field theory, the boundary problem requires explicit bosonic and fermionic boundary conditions, and those choices generate distinct surface universality classes with different boundary spectra, stability properties, and free energies. Current work organizes these classes into normal, ordinary, and special phases, while semi-infinite renormalization-group analyses also resolve time-reversal-invariant (TRI) and non-TRI fermionic boundary conditions. The expression “boundary Gross–Neveu–Yukawa fixed point” is used nonuniformly: some analyses apply it broadly to interacting surface fixed points built on the bulk GNY critical point, whereas higher-order AdS and large-\(N\) work uses it more specifically for the normal class [2603.07637, 2606.07510, 2110.04268].

## 1. Semi-infinite and AdS formulations

In the semi-infinite formulation, Euclidean spacetime is \(D=d+1\) dimensional with coordinates
\[
\bm{x}=(x_0,\dots,x_d)\equiv (\vec r,z),\qquad z\ge 0,
\]
so the boundary is the plane \(z=0\). The bulk action is the GNY theory of Dirac fermions \(\psi_a\) coupled to a real scalar \(\varphi\),
\[
S=S_{\mathrm{GNY}}+S_{\mathrm B},
\]
with
\[
S_{\mathrm{GNY}}=\int d^dr\int_0^\infty dz\, \Bigg[ \sum_{a=1}^{\tilde N} \bar\psi_a(\slashed\partial+g\varphi)\psi_a +\frac12(\nabla\varphi)^2 +\frac12 m^2\varphi^2 +\frac{\lambda}{4!}\varphi^4 \Bigg],
\]
and boundary terms
\[
S_{\mathrm B} =\int d^dr\left[ \sum_a \bar\psi_a \check M \psi_a +\frac{c}{2}\varphi^2 \right].
\]
Here \(g\) is the Yukawa coupling, \(\lambda\) the bosonic self-coupling, \(m^2\) the bulk mass tuning parameter, and \(N=\tilde N\,\mathrm{tr}\,\mathbb 1\) the effective fermion flavor factor [2603.07637].

The AdS formulation is Weyl-equivalent to the flat-space BCFT and uses
\[
ds^2=\frac{dz^2+d\bold{x}^2}{z^2},\qquad z\ge 0.
\]
Near \(d=4\), the same universality class is written as
\[
S_{\rm GNY}=\int d^d x \left[\frac{1}{2}(\partial_\mu s)^2 -\left(\bar\Psi^I \slashed{\partial}\Psi_I+g_1 s\,\bar\Psi^I\Psi_I\right) +\frac{g_2}{4!}s^4\right], \qquad I=1,\dots,N.
\]
This AdS description is central in the large-\(N\) analysis of boundary conformal phases and in comparisons between \(4-\epsilon\) and \(1/N\) expansions [2110.04268, 2606.07510].

## 2. Boundary conditions and symmetry constraints

For the scalar field, the natural boundary term produces a Robin condition
\[
\partial_z\varphi\big|_{z=0}=c\,\varphi\big|_{z=0}.
\]
Two RG-relevant limits are singled out. The special transition corresponds to \(c=c^*_{\rm sp}=0\) in minimal subtraction and gives Neumann boundary conditions,
\[
\partial_z\varphi|_{z=0}=0,
\]
while the ordinary transition corresponds to \(c=c^*_{\rm ord}=\infty\) and gives Dirichlet boundary conditions,
\[
\varphi|_{z=0}=0.
\]
Using
\[
\chi= \begin{cases} +1 & \text{Neumann / special},\\ -1 & \text{Dirichlet / ordinary}, \end{cases}
\]
the free boson propagator takes the form
\[
G^{\chi}_{\mathrm B}(\vec q,z,z') =\frac{1}{2\kappa}\Big[e^{-\kappa|z-z'|}+\chi\, e^{-\kappa(z+z')}\Big], \qquad \kappa=\sqrt{q^2+m^2}
\]
[2603.07637].

The fermionic boundary condition is imposed by a projector,
\[
M\,\psi|_{z=0}=\psi|_{z=0},
\]
where \(M\) anticommutes with the normal Dirac matrix \(\alpha_d\) so that the normal current vanishes, \(J_d=\psi^\dagger \alpha_d \psi=0\). The most general family compatible with unitarity, conformal invariance, and charge conjugation symmetry is
\[
M=-(\gamma_d\sin\phi+i\gamma_d\gamma_S\cos\phi),\qquad \check M=\sin\phi+i\gamma_S\cos\phi,
\]
with \(\phi\in(-\pi,\pi]\). In \(D=3\) and \(D=4\), this exhausts the symmetry-allowed boundary conditions. The special TRI choices are
\[
\phi=\pm \frac{\pi}{2},
\]
whereas
\[
\phi=0,\pi
\]
are non-TRI fixed points. For the symmetry-allowed family \(M(\phi)\), there are no localized boundary states in the free fermion problem; the would-be surface mode merges into the bulk continuum [2603.07637].

## 3. Boundary universality classes and phase structure

The large-\(N\) and AdS analyses identify three conformal boundary universality classes for the GNY/Gross–Neveu system [2110.04268, 2606.07510].

| Class | Scalar sector | Boundary characterization |
|---|---|---|
| Normal | \(s_0=s_{\star,0}+t\), \(s_{\star,0}=\sqrt{\frac{3d(d-2)}{2g_{2,0}}}\) | analogue of extraordinary/normal; induces fermion mass |
| Ordinary | \(s_0=0\) | Dirichlet boundary condition for the scalar |
| Special | \(s_0=0\) | Neumann boundary condition for the scalar |

In the normal phase, the nontrivial scalar saddle induces
\[
\mu_0=g_{1,0}s_{\star,0}.
\]
Near \(d=4\), only the standard fermion quantization is unitary in this phase, and the higher-order AdS analysis identifies this phase as the boundary Gross–Neveu–Yukawa fixed point in the narrow sense [2606.07510]. In the earlier large-\(N\) Gross–Neveu BCFT language, the same phase is denoted \(B_1\), while the Dirichlet-like and Neumann-like phases are \(B_2\) and \(B_2'\), with the correspondence
\[
B_2' \leftrightarrow \text{Neumann},\qquad B_2 \leftrightarrow \text{Dirichlet},\qquad B_1 \leftrightarrow \text{vev phase}
\]
[2110.04268].

The RG hierarchy between these phases is
\[
B_2' \longrightarrow B_2 \longrightarrow B_1.
\]
The deformation \(B_2'\to B_2\) is generated by the boundary operator \(\hat s^2\), while \(B_2\to B_1\) is generated by the linear boundary operator \(\hat s\). In this ordering, \(B_1\) is the most stable phase under boundary RG flow [2110.04268].

The semi-infinite \(4-\epsilon\) analysis refines this picture by organizing flows in the \((c,\phi)\) plane. It identifies ordinary, special, and extraordinary bosonic sectors together with TRI and non-TRI fermionic sectors. The ordinary phase is stable for \(c>c^*_{\rm sp}\), the extraordinary phase is stable for \(c<c^*_{\rm sp}\), and the special phase is unstable. On the fermionic side, TRI fixed points are unstable under TRI-breaking boundary perturbations, whereas non-TRI fixed points are stable [2603.07637]. This implies that the phrase “boundary GNY fixed point” requires care: one usage isolates the normal phase, while another refers more broadly to the interacting boundary fixed points obtained by combining the bulk GNY critical point with admissible surface boundary conditions.

## 4. Renormalization-group structure and one-loop boundary exponents

In dimensional regularization and minimal subtraction with \(D=4-\varepsilon\), the bulk beta functions are
\[
\beta_{g^2}=-\varepsilon g^2+(N+6)g^4,
\]
\[
\beta_\lambda=-\varepsilon\lambda+3\lambda^2+2N\lambda g^2-12Ng^4.
\]
The interacting bulk fixed point is
\[
g^{*2}=\frac{\varepsilon}{N+6},\qquad \lambda^*=\frac{Y(N)-N+6}{6(N+6)}\varepsilon,
\]
with
\[
Y(N)=\sqrt{N^2+132N+36}.
\]
This GNY fixed point is IR stable for \(D<4\) [2603.07637].

At one loop, the bulk anomalous dimensions are
\[
\eta_\psi=\frac{\varepsilon}{N+6},\qquad \eta_\varphi=\frac{N\varepsilon}{N+6},
\]
and
\[
\frac{1}{\nu} =2-\frac{Y(N)+5N+6}{6(N+6)}\varepsilon.
\]
The boundary sector depends on both the bosonic choice \(\chi=\pm1\) and the fermionic label
\[
\sigma= \begin{cases} +1 & \text{TRI},\\ -1 & \text{non-TRI}. \end{cases}
\]

For the boundary fermion,
\[
\eta_{\psi_\parallel} =-\left[(2+\chi)\sigma+N\frac{\sigma+1}{2}\right]\frac{\varepsilon}{N+6},
\]
and
\[
\Delta_{\psi_s} =\frac{3}{2} -\frac{(\sigma+3)N+2(2+\chi)\sigma+12}{4(N+6)}\varepsilon.
\]
At the special transition,
\[
\eta_{\varphi_\parallel} =-\frac{Y(N)+(6\sigma-1)N+6}{6(N+6)}\,\varepsilon,
\]
\[
\Delta_{\varphi_s} =1-\frac{Y(N)+(6\sigma+5)N+42}{12(N+6)}\,\varepsilon.
\]
At the ordinary transition, where the natural renormalized operator is \(\partial_z\varphi|_{z=0^+}\),
\[
\eta_{\varphi_\parallel}^{\rm(ord)} =2-\frac{Y(N)-(6\sigma+1)N+6}{6(N+6)}\,\varepsilon,
\]
\[
\Delta_{\varphi_s} =2-\frac{Y(N)-(6\sigma-5)N+42}{12(N+6)}\,\varepsilon.
\]
The crossover exponent at the special transition is
\[
\Phi=\frac12-\frac{Y(N)-(12\sigma+7)N+6}{24(N+6)}\,\varepsilon.
\]
These expressions show explicitly that boundary critical behavior is not fixed by the bulk universality class alone; it depends on the surface scalar sector and on the fermionic boundary projector [2603.07637].

## 5. Large-\(N\), AdS quantization, and boundary operator spectra

In AdS, a free massive fermion admits two boundary quantizations,
\[
\gamma^0\Psi(z\to 0,\bold{x})=\mp \operatorname{sign}(m)\,\Psi(z\to 0,\bold{x}),
\]
with boundary dimensions
\[
\hat\Delta_\pm=\frac{d-1}{2}\pm |m|.
\]
The alternative quantization is unitary only for \(|m|<1/2\), which is why near \(d=4\) the normal phase retains only the standard quantization [2110.04268, 2606.07510].

At leading large \(N\), the normal phase has
\[
\hat\Delta_{(1/2)}^{(0)}=d-\frac{3}{2}.
\]
The first \(1/N\) correction is
\[
\hat{\Delta}_{(1/2)} = d-\frac{3}{2} +\frac{1}{N}\, \frac{2^{2d-3}\Gamma\!\left(d-\frac12\right)} {\sqrt{\pi}\,c_d\,d\,\Gamma(d-2)} +O\!\left(\frac{1}{N^2}\right),
\]
with \(c_d=2^{d/2}\) in the conventions of the calculation. In \(d=3\),
\[
\hat\gamma_\Psi=\frac{1}{N},\qquad \hat\Delta_{(1/2)}=\frac{3}{2}+\frac{1}{N}+O(1/N^2).
\]
The normal-phase scalar boundary spectrum contains operators of dimension
\[
\hat\Delta = d+2n,\qquad n=0,1,2,\dots,
\]
with the leading operator being the displacement operator. Consequently, there is no relevant \(O(N)\)-invariant boundary perturbation at this fixed point at leading large \(N\) [2606.07510].

By contrast, the ordinary and special phases are distinguished by the leading scalar boundary operator. In the Dirichlet-like phase its leading dimension is \(2\), while in the Neumann-like phase it is \(d-3\). Near \(d=4\), the corresponding scalar dimensions in the GNY description are
\[
\hat\Delta_s^N = 1 - \frac{\sqrt{4N^2+132N+9}+22N+21}{12(2N+3)}\epsilon,
\]
\[
\hat\Delta_s^D = 2 - \frac{\sqrt{4N^2+132N+9}-2N+21}{12(2N+3)}\epsilon.
\]
The fermion boundary dimensions in these two phases are
\[
\hat\Delta_{\Psi}^{N} = \frac{3}{2} - \frac{8N+9}{4(2N+3)}\epsilon, \qquad
\hat\Delta_{\Psi}^{D} = \frac{3}{2} - \frac{8N+7}{4(2N+3)}\epsilon.
\]
This spectrum underlies the RG ordering \(B_2'\to B_2\to B_1\): the Neumann-like phase has the most relevant scalar boundary operator, the Dirichlet-like phase has a less relevant one, and the normal phase has none [2110.04268].

## 6. Boundary free energy and boundary central charge

AdS methods make the boundary free energy calculable beyond leading order. For the normal phase, the first subleading correction \(F_{\rm nor}^{(1)}\) was obtained by harmonic analysis, and in \(d=3\) it reduces to
\[
F_{\rm nor}^{(1)}=\frac{3}{16}\log R.
\]
This gives the large-\(N\) boundary central charge
\[
c_{\rm nor}=-\frac{N}{4}-\frac{9}{16}+O(1/N).
\]
For the ordinary phase,
\[
F_{\rm ord}^{(1)} = F_D-\frac{3}{32}\epsilon+O(\epsilon^2),
\]
and in \(d=3\),
\[
F_{\rm ord}^{(1)}=\frac{\log R}{48}.
\]
For the special phase,
\[
F_{\rm spec}^{(1)}=-\frac{\epsilon}{16}-\frac{61}{576}\epsilon^2+O(\epsilon^3).
\]
The agreement of these quantities between large \(N\) and \(4-\epsilon\) expansions is a central consistency check of the phase identification [2606.07510].

The boundary \(c\)-function is introduced through
\[
\tilde s = -\sin\!\left(\frac{\pi(d-1)}{2}\right)\left(F-\frac12 F_{S^d}\right),
\]
and in \(d=3\),
\[
\tilde s = \frac{\pi}{6} c_{\rm bdry}.
\]
Using the \(\epsilon\)-expanded AdS free energy and Padé resummation, the normal phase at the supersymmetric point \(N=\frac14\) yields
\[
c_{\rm bdry}^{\rm nor}\approx -1.38,
\qquad
a_s^{\rm nor}(d=3)\approx 2.69,
\]
while the ordinary phase gives
\[
c_{\rm bdry}^{\rm ord}\approx -0.015.
\]
Earlier large-\(N\) free-energy comparisons also showed that the rescaled boundary free energy decreases along the RG flows \(B_2'\to B_2\to B_1\), in accord with the boundary \(F\)-theorem [2606.07510, 2110.04268].

## 7. Relation to neighboring models and to bulk GNY studies

The semi-infinite pseudoscalar Yukawa model is bulk-equivalent but not boundary-equivalent to the GNY model. In the bulk, the chiral rotation
\[
\psi\to e^{-i\pi\gamma_S/4}\psi,\qquad \bar\psi\to\bar\psi e^{-i\pi\gamma_S/4}
\]
maps the pseudoscalar Yukawa model to GNY, but on the half-space the boundary matrix transforms as
\[
\check M\to e^{-i\pi\gamma_S/4}\check M e^{-i\pi\gamma_S/4},
\]
which swaps TRI and non-TRI boundary conditions, for example
\[
\check M=\pm \mathbb 1 \longleftrightarrow \check M=\mp i\gamma_S.
\]
This explains why the semi-infinite pseudoscalar Yukawa model does not share the same boundary critical exponents as the semi-infinite GNY model [2603.07637].

A separate source of confusion is the distinction between boundary and bulk GNY criticality. The mixed-correlator bootstrap study of the \(3d\) \(O(N)\)-symmetric GNY fixed points constrains bulk scaling dimensions and isolates small “islands” for \(\Delta_\psi\), \(\Delta_\sigma\), and \(\Delta_\epsilon\), but it does not analyze boundary operator spectra, boundary conditions, or defect crossing equations. Its connection to boundary physics is indirect, such as the observation that the \(N=1\) bulk GNY model is expected to describe a transition at the boundary of a topological superconductor; the bootstrap itself remains a bulk-CFT analysis [2210.02492].

Taken together, these developments place the boundary Gross–Neveu–Yukawa problem within the standard framework of boundary critical phenomena while adding genuinely fermionic structure. The scalar boundary condition selects ordinary, special, or normal behavior; the fermionic projector further resolves TRI and non-TRI sectors; and the resulting BCFT data can be tracked in parallel by \(4-\epsilon\), large-\(N\), and AdS methods. The remaining ambiguity in the phrase “boundary GNY fixed point” reflects the coexistence of a broad semi-infinite RG usage and a narrower AdS/large-\(N\) usage, rather than a single universally adopted nomenclature.

Source: https://www.emergentmind.com/topics/boundary-gross-neveu-yukawa-fixed-point