A Complex Fermionic Tensor Model in $d$ Dimensions

TL;DR

This work analyzes a melonic fermionic tensor model in $d$ dimensions with three-index Dirac fermions and a four-fermion interaction in the large-$N$ limit. By summing melonic diagrams at strong coupling, the authors obtain a formal conformal fixed point across dimensions, determine the fermion scaling dimension $Δ_ψ=rac{d}{4}$, and compute the spectrum of scalar bilinear operators via an integral kernel arising from a Schwinger-Dyson framework. Numerically, the spectrum is real for $d<2$ but develops complex eigenvalues for $2<d<6$, signaling instability, while a narrow real window appears near $d=6$ in a $6+oldsymbol{ ext{ε}}$ expansion; analytic $oldsymbol{ ext{ε}}$-expansions around $d=2$ and $d=6$ provide explicit scaling-dimension series. The results offer a potential route to study 1D tensor models using an $oldsymbol{ ext{ε}}$-expansion and shed light on possible UV fixed points and holographic interpretations, albeit with unitarity considerations when restricting to singlet sectors.

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 $d=2-ε$ 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 $ε$. The theory appears to be weakly interacting when $ε$ is small, suggesting that fermionic tensor models in 1-dimension can be studied in an $ε$ 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 $ε$ in $6+ε$ dimensions.

Figures (11)

• Figure 1: The Schwinger-Dyson Equation for the exact propagator.
• Figure 2: The Schwinger-Dyson Equation for the exact three point function $\langle \psi(x_1) O_s(x_3,\epsilon_3) \bar{\psi}(x_2)\rangle$.
• Figure 3: Feynman diagrams corresponding to melonic Wick contractions proportional to $\lambda_1^2$ or $\lambda_2^2$. The above diagram shows contraction of spinor indices, and the lower diagram shows contraction of colour indices.
• Figure 4: Feynman diagrams corresponding to melonic Wick contractions proportional to $\lambda_1 \lambda_2$. The above diagram shows contraction of spinor indices, and the lower diagram shows contraction of colour indices.
• Figure 5: A plot $g_{\text{odd}}(3,\tau)$ and $1$ for $d=3$.