A Supersymmetric SYK-like Tensor Model

TL;DR

Without quenched disorder, the authors construct a supersymmetric SYK-like tensor model with quark and meson superfields and establish a well-defined large-$N$ limit in which meson melons dominate. The Schwinger-Dyson equations in the IR reproduce the FGMS results, yielding IR dimensions $ ext{Δ}_ ext{ψ}= ext{Δ}_ ext{χ}= frac{1}{6}$ and $ ext{Δ}_ ext{β}= ext{Δ}_ ext{π}= frac{2}{3}$, and the 4-point kernel matches FGMS, implying the same operator spectrum and chaotic behavior. The construction thus provides a tensor-version of FGMS and demonstrates how disorder-free SUSY tensor theories can emulate SYK-like IR physics. It also highlights the sensitivity of large-$N$ behavior to the order of integrating out auxiliary fields and points to future explorations of purely mesonic sectors and higher-dimensional generalizations.

Abstract

We consider a supersymmetric SYK-like model without quenched disorder that is built by coupling two kinds of fermionic N=1 tensor-valued superfields, "quarks" and "mesons". We prove that the model has a well-defined large-N limit in which the (s)quark 2-point functions are dominated by mesonic "melon" diagrams. We sum these diagrams to obtain the Schwinger-Dyson equations and show that in the IR, the solution agrees with that of the supersymmetric SYK model.

Figures (19)

• Figure 1: The large-$N$ limit of the SYK-like model of Witten:2016iux, like many tensor models, is dominated by "melon" graphs. Here we show several melonic contributions to the 2-point function. In general they may be generated iteratively using the basic building block shown in the second term.
• Figure 2: Left: the dominant "core" diagram contributing to the $\psi\psi$ 2-point function in the supersymmetric FGMS model. Right: the corresponding dominant "core" diagram in the fermionic SYK model. In both graphs the solid lines represent $\psi_i$ fields, the wavy lines represent $b_i$ fields, and the dashed lines represent a correlation of a product of Gaussian random couplings; $C_{ijk}$ on the left and $J_{ijkl}$ on the right. The full set of dominant graphs is generated iteratively using this core.
• Figure 3: Left: the dominant "core" diagram contributing to the $bb$ 2-point function in the supersymmetric FGMS model. Right: the dominant ladder "rung" diagram contributing to the $\psi\psi\psi\psi$ 4-point function in the fermionic SYK model. The full set of dominant graphs is generated iteratively using this core (with propagator corrections obtained by also iterating the right panel of figure \ref{['fig:susy2pt']}).
• Figure 4: The first few diagrams which dominate the $\psi\psi$ (top panel) and $\chi\chi$ (bottom panel) 2-point functions (and their super-descendants) in the large-$n$ limit when $g \sim 1/n$. In each panel the five diagrams scale as $g^k n^k$ for $k=0, 2, 4, 4, 4$, respectively. As noted in the text, single lines represent $\psi_a$ or $\beta_a$ fields and double lines represent the mesonic $\chi_{ab}$ or $\pi_{ab}$ fields. All dominant diagrams are obtained by iterating the building blocks shown in figure \ref{['fig:susy4ptwithbos']}.
• Figure 5: Fundamental building blocks for the $\psi\psi$ (left) and $\chi\chi$ (right) 2-point functions (and their super-descendants) at large $n$. Each of these two diagrams scales as $g^2n^2$.