Chaotic RG Flow in Tensor Models

TL;DR

This paper investigates chaotic behavior in the RG flow of a bi-antisymmetric tensor quantum field theory with $O(N_1) imes O(N_2)$ symmetry, revealing chaos near bifurcation points through analytical and numerical methods.

Contribution

It introduces the analysis of chaotic RG flows in tensor models, identifying bifurcation points and demonstrating chaos via Shilnikov orbits and complex coupling behaviors.

Findings

Chaotic RG flow occurs near zero-Hopf bifurcation points.

Analytical and numerical evidence of Shilnikov homoclinic orbits.

Chaotic behavior also observed in a non-hermitian Ising chain with complex couplings.

Abstract

We study a bi-antisymmetric tensor quantum field theory with $O(N_1)\times O(N_2)$ symmetry. Working in $4-\epsilon$ dimensions we calculate the beta functions up to second order in the coupling constants and analyze in detail the Renormalization Group (RG) flow and its fixed points. We allow $N_1$ and $N_2$ to assume general real values and treat them as bifurcation parameters. In studying the behavior of the model in the space of $N_1$ and $N_2$ we find a point where a zero-Hopf bifurcation occurs. In the vicinity of this point, we provide analytical and numerical evidence for the existence of Shilnikov homoclinic orbits, which induce chaotic behavior in the RG flow of the model. As a simple warm-up example for the study of chaotic RG flows, we also review the non-hermitian Ising chain and show how for special complex values of the coupling constant, its RG transformations are chaotic and equivalent to the Bernoulli map.