A Complex Fermionic Tensor Model in $d$ Dimensions

TL;DR

This paper investigates a fermionic tensor model in arbitrary dimensions, analyzing its spectrum and fixed points, revealing stability in certain dimensions and potential instability in others, with implications for understanding fermionic theories in various dimensions.

Contribution

It introduces a melonic fermionic tensor model in arbitrary dimensions and computes its spectrum and fixed points, extending the analysis of tensor models beyond known cases.

Findings

In $d=2- ext{epsilon}$, the model has a real spectrum and is an infrared fixed point.

For $2<d<6$, the spectrum includes complex eigenvalues indicating instability.

In $6<d<6.14$, the fixed point is weakly interacting with a real spectrum.

Abstract

In this note, we study a melonic tensor model in $d$ dimensions based on three-index Dirac fermions with a four-fermion interaction. Summing the melonic diagrams at strong coupling allows one to define a formal large-$N$ saddle point in arbitrary $d$ and calculate the spectrum of scalar bilinear singlet operators. For $d=2-\epsilon$ the theory is an infrared fixed point, which we find has a purely real spectrum that we determine numerically for arbitrary $d<2$, and analytically as a power series in $\epsilon$. The theory appears to be weakly interacting when $\epsilon$ is small, suggesting that fermionic tensor models in 1-dimension can be studied in an $\epsilon$ expansion. For $d>2$, the spectrum can still be calculated using the saddle point equations, which may define a formal large-$N$ ultraviolet fixed point analogous to the Gross-Neveu model in $d>2$. For $2<d<6$, we find that the spectrum contains at least one complex scalar eigenvalue (similar to the complex eigenvalue present in the bosonic tensor model recently studied by Giombi, Klebanov and Tarnopolsky) which indicates that the theory is unstable. We also find that the fixed point is weakly-interacting when $d=6$ (or more generally $d=4n+2$) and has a real spectrum for $6<d<6.14$ which we present as a power series in $\epsilon$ in $6+\epsilon$ dimensions.