Vector models and generalized SYK models

TL;DR

The paper investigates how SYK-like chaos can emerge from weakly coupled vector dynamics by constructing tensor–vector toy models. Through large-$N$ analysis, Schwinger-Dyson equations, and ladder/retarded kernels, it shows that integrating out a tensor yields a 1D Gross-Neveu theory, while certain marginal perturbations drive the IR to chaotic SYK-like fixed points with a maximal Lyapunov exponent $\lambda_L = 2\pi/\beta$. A phase transition controlled by the perturbation sign separates chaotic and nonchaotic phases, and several symmetry variants consistently exhibit chaos without disorder. The work highlights a structural bridge between GN-like vector models and SYK chaos, with potential holographic interpretations in AdS$_2$ or dilaton-gravity contexts and implications for organizing operator towers in chaotic sectors.

Abstract

We consider the relation between SYK-like models and vector models by studying a toy model where a tensor field is coupled with a vector field. By integrating out the tensor field, the toy model reduces to the Gross-Neveu model in 1 dimension. On the other hand, a certain perturbation can be turned on and the toy model flows to an SYK-like model at low energy. A chaotic-nonchaotic phase transition occurs as the sign of the perturbation is altered. We further study similar models that possess chaos and enhanced reparameterization symmetries.

Figures (12)

• Figure 1: The fundamental melon that dominates all the corrections to the two point function of the $\chi^i$ field. The wavy lines in the diagram represent tensor field, and the solid lines represent the vector fermion. Direction of the arrows distinguishes a field from its conjugation; any line with its arrow going into (out of) a vertex represents a (conjugate) field.
• Figure 2: The ladder diagrams that dominate the 4-point correction functions of the vector fields. The solid and wavy lines represent the vector and tensor field respectively.
• Figure 3: The effective coupling of the vector model obtained by integrating out the tensor fields. Notice that a similar 6-point effective coupling is not generated due to the form of the interaction \ref{['action']}.
• Figure 4: Mapping the melonic diagrams of the model \ref{['action']} onto the bubble diagrams of the vector model in the limit $J\ll M$. Here we have assumed a (small) mass term for the vector fermion. If this is not the case, the snail diagrams in the vector model and the corresponding diagrams in the model \ref{['action']} vanish.
• Figure 5: Schematic form of the eigen-equations associated to the kernels of the 4-point functions in the model \ref{['H1']}. The solid, dashed and wavy lines represent the vector fermions, the vector bosons and the tensor fields respectively. The letters in the blobs of the eigenvectors indicate whether it is the symmetric or antisymmetric eigenfunction.