Bootstrapping 3D Fermions

TL;DR

This work extends the conformal bootstrap to 4-point functions of 3D fermions by constructing an embedding-space formalism that expresses fermionic conformal blocks as derivatives of scalar blocks. It derives general bounds on the dimensions of operators in the $\psi\times\psi$ OPE and on the central charge $C_T$, uncovering sharp features such as a kink at $\Delta_ψ\approx1.27$ that suggests a possible dead-end CFT with no relevant scalars. Imposing gaps in the scalar spectrum reveals kinks aligning with large-$N$ Gross-Neveu models, mapping out how GN-like fixed points appear as the second parity-odd scalar gap is varied, and highlighting a second robust feature near $(\Delta_ψ,\Delta_σ)\approx(1.078,0.565)$. The results point to rich fermionic CFT structure in 3D and motivate future work with mixed correlators to isolate specific theories, including the ${\cal N}=1$ SUSY Ising model and $O(N)$-symmetric extensions.

Abstract

We study the conformal bootstrap for a 4-point function of fermions $\langleψψψψ\rangle$ in 3D. We first introduce an embedding formalism for 3D spinors and compute the conformal blocks appearing in fermion 4-point functions. Using these results, we find general bounds on the dimensions of operators appearing in the $ψ\times ψ$ OPE, and also on the central charge $C_T$. We observe features in our bounds that coincide with scaling dimensions in the Gross-Neveu models at large $N$. We also speculate that other features could coincide with a fermionic CFT containing no relevant scalar operators.

Figures (8)

• Figure 1: Upper bounds on the dimension of the lowest dimension parity-odd scalar appearing in the $\psi \times \psi$ OPE, assuming only conformal symmetry, parity symmetry, and unitarity. The orange region is allowed, and the white region is disallowed. The black dashed line starting at the free theory point $(\Delta_{\psi}, \Delta_\sigma) = (1, 2)$ gives the relation among dimensions specific to Mean Field Theory, while the dashed line starting at $(\Delta_{\psi}, \Delta_\sigma) = (1, 0.5)$ gives the relation among dimensions expected for ${\cal N} = 1$ SCFTs, assuming $\psi$ is a superdescendant of $\sigma$. These bounds are determined using the procedure described in Section \ref{['sec:bootstrap']} (see also Appendix \ref{['app:sdpb']}) by performing a binary search in $\Delta_{\sigma}$ with $10^{-3}$ precision. The parameter $\Lambda$ defined in Appendix \ref{['app:sdpb']} is given by $\Lambda = 23$.
• Figure 2: Upper bound on the lowest dimension parity-even scalar appearing in the $\psi \times \psi$ OPE, as a function of $\Delta_\psi$, assuming only conformal symmetry, parity symmetry, and unitarity. As $\Delta_\psi \to 1$, the bound goes to $\Delta_\epsilon =3$ and has asymptotic behavior $\Delta_\epsilon-3\propto(\Delta_\psi-1)^{1/2}$. The bound has a kink at $\Delta_\psi = 1.27$, which is the same value of $\Delta_\psi$ at which the bound for parity-odd scalars had a discontinuity, see Figure \ref{['fig:noRelevantParityOdd']}. This bound was computed with $\Lambda=23$.
• Figure 3: Allowed values of the dimensions $(\Delta_\psi,\Delta_\sigma)$, assuming $\Delta_{\sigma'}\geq \Delta_{\sigma'}^\mathrm{min}$ for $\Delta_{\sigma'}^\mathrm{min}\in\{2.01,2.03,2.05,2.07,2.09,2.11\}$, computed with $\Lambda=19$. The regions to the right of their respective curves (shaded orange) are allowed, while the regions to the left are disallowed. The black dashed line shows the relationship between $\Delta_{\psi}$ and $\Delta_\sigma$ using the known 2-loop (for $\Delta_\sigma$) and 3-loop (for $\Delta_\psi$) large-$N$ results in Table \ref{['tab:grossNeveuDimensions']} and Appendix \ref{['GNYAPPENDIX']}. The free theory at $(\Delta_\psi,\Delta_\sigma)=(1,2)$ is always allowed. Below the free theory, there are kinks that closely track the dimensions of operators in the Gross-Neveu models at large $N$. The vertical lines at the bottom of the first two curves ensure consistency of the bounds with Mean Field Theory.
• Figure 4: Allowed values of the dimensions $(\Delta_\psi,\Delta_\sigma)$, assuming $\Delta_{\sigma'}\geq \Delta_{\sigma'}^\mathrm{min}$ for $\Delta_{\sigma'}^\mathrm{min}\in\{2.1,2.3,2.5,2.7,2.9\}$, computed with $\Lambda=19$. The regions to the right of their respective curves (shaded orange) are allowed, while the regions to the left are disallowed. The black dashed line shows the relationship between $\Delta_{\psi}$ and $\Delta_\sigma$ at 2- and 3-loops at large-$N$.
• Figure 5: The positions of the kinks in Figure \ref{['fig:grossNeveuSmallSigPrime']} (black points), compared with the 1-loop large-$N$ prediction $\Delta_{\sigma'}=8\Delta_\psi-6$ for the 3D Gross-Neveu models in Table \ref{['tab:grossNeveuDimensions']} (orange line). We also indicate the approximate value of $N$ corresponding to each kink.