Bootstrapping 3D Fermions with Global Symmetries

TL;DR

The paper advances the 3d conformal bootstrap by analyzing fermionic four-point functions ⟨ψ_i ψ_j ψ_k ψ_ℓ⟩ with O($N$) symmetry, using semidefinite programming to bound scaling dimensions and central charges. It uncovers a sequence of kinks in the bounds that align with Gross-Neveu–Yukawa fixed points at large $N$ and provides nonperturbative predictions for small $N$, including hints of a second class of CFTs (GNY$^*$) under certain gap assumptions. The results are cross-validated against large-$N$ and ε-expansion analyses, and they offer insights into the operator spectrum ($Δ_ψ$, $Δ_{σ}$, $Δ_{σ_T}$, $Δ_{ε}$) and central charges ($C_J$, $C_T$). The work also highlights the potential existence of dead-end CFTs and outlines future directions such as mixed correlators, higher-derivative bounds, parity-violating cases, and extensions to other spacetime dimensions, which could impact the understanding of quantum criticality in condensed-matter systems.

Abstract

We study the conformal bootstrap for 4-point functions of fermions $\langle ψ_i ψ_j ψ_k ψ_{\ell} \rangle$ in parity-preserving 3d CFTs, where $ψ_i$ transforms as a vector under an $O(N)$ global symmetry. We compute bounds on scaling dimensions and central charges, finding features in our bounds that appear to coincide with the $O(N)$ symmetric Gross-Neveu-Yukawa fixed points. Our computations are in perfect agreement with the $1/N$ expansion at large $N$ and allow us to make nontrivial predictions at small $N$. For values of $N$ for which the Gross-Neveu-Yukawa universality classes are relevant to condensed-matter systems, we compare our results to previous analytic and numerical results.

Figures (7)

• Figure 1: Upper bounds on the scaling dimension of the lowest-lying parity-odd $O(N)$ symmetric-traceless tensor, $\Delta_{\sigma_T}$, for a unitary CFT containing fermions with scaling dimension $\Delta_\psi$. We focus on the cases $N=2,\, 3,\,4, \, 10$ and $20$. The black symbols show the estimated values of the scaling dimensions $\Delta_{\psi}$ and $\Delta_{\sigma_T}$ obtained from the large-$N$ expansion up to $O(1/N^4)$ corrections fei2016yukawa. For $N=10,\, 20$ we note strong agreement. This figure is a zoomed in version of Figure \ref{['fig:univ-bounds-sigmaT']}.
• Figure 2: Bound on the scaling dimension of the lowest-lying parity-odd $O(N)$ singlet, $\Delta_{\sigma}$, for a unitary CFT containing fermions with scaling dimension $\Delta_\psi$, when imposing a gap above this operator up to $\Delta_{\sigma'} > 3$. Once again, we focus on the cases $N=2,\, 3,\,4,$ and $10$. We notice that when imposing such a gap, we observe features that are close to the values of $\Delta_{\psi}$ found for the GNY kinks in Figure \ref{['fig:univ-bounds-sigmaT-zoomIn']}. Estimates from the large-$N$ expansion (black markers) for the scaling dimension of $\Delta_{\sigma}$ also agree well with the position of the kink for $N=10$, but is inaccurate at smaller values of $N$. The red markers are the three-loop $\epsilon$-expansion results for the dimensions of the GNY models after performing a Pade$_{[2,1]}$ approximation (see Eqs. (11)--(13) and Table II in 2017arXiv170308801M). They are reasonably close to the upper kinks in the bounds for small $N$. The lower kinks appear close to the three-loop $\epsilon$-expansion estimates for the GNY$^*$ models (green hollow markers), obtained after performing a Pade$_{[1,2]}$ approximation, following the methods in fei2016yukawa and 2017arXiv170308801M. While these second kinks are close to the $\epsilon$-expansion estimates for small $N$, for $N=10$ the second kink does not exist at all.
• Figure 3: Upper bounds on the scaling dimension of the lowest-lying parity-odd $O(N)$ singlet, $\Delta_{\sigma}$, for a unitary CFT containing fermions with dimension $\Delta_\psi$. Above, we focus on the cases $N=2,\, 3,\,4, \, 10$ and $20$. As $\Delta_\psi \to 1$, the bound approaches the free theory value of $\Delta_{\sigma} = 2$. Reminiscent of the jump noticed in our previous work Iliesiu:2015qra when bounding the scaling dimension of the lowest-lying parity-odd operator in a theory with one fermion, we observe a different jump for each value of $N$ that we study. Once again, all jumps occur once the bound intersects the horizontal line on which $\sigma$ is precisely marginal, $\Delta_{\sigma} = 3$.
• Figure 4: Upper bounds on the scaling dimension of the lowest-lying parity-even $O(N)$ singlet, $\Delta_{\epsilon}$, for a unitary CFT containing fermions with scaling dimension $\Delta_\psi$. Once again, we focus on the cases $N=2,\, 3,\,4, \, 10$ and $20$. As $\Delta_\psi \to 1$, the bound approaches the value of $\Delta_{\epsilon} = 3$.
• Figure 5: Upper bounds on the scaling dimension of the lowest-lying parity-odd $O(N)$ symmetric-traceless tensor, $\Delta_{\sigma_T}$, for a unitary CFT containing fermions with scaling dimension $\Delta_\psi$. Once again, we focus on the cases $N=2,\, 3,\,4, \, 10$ and $20$. As $\Delta_\psi \to 1$, the bound approaches the free theory value of $\Delta_{\sigma_T} = 3$. There are two sets of kinks. The first set of kinks correspond to the GNY-models and are located in the square on the lower left. We zoom onto this lower left square in Figure \ref{['fig:univ-bounds-sigmaT-zoomIn']} and discuss the meaning of the kinks in more detail in Section \ref{['sec:GNY_spectrum']}.