Growth of a renormalized operator as a probe of chaos

TL;DR

The paper suggests that the growth of a renormalized operator under holographic RG flow indicates chaos saturation, supported by toy models showing linear growth of operator size.

Contribution

It introduces a new perspective linking operator growth to chaos saturation and tests this idea using tensor network and error-correcting code models.

Findings

Operator size grows linearly in toy models.

Supports the conjecture relating operator growth to chaos bound saturation.

Uses both MERA-like tensor networks and perfect tensor codes.

Abstract

We propose that the size of an operator evolved under holographic renormalization group flow shall grow linearly with the scale and interpret this behavior as a manifestation of the saturation of the chaos bound. To test this conjecture, we study the operator growth in two different toy models. The first is a MERA-like tensor network built from a random unitary circuit with the operator size defined using the integrated out-of-time-ordered correlator (OTOC). The second model is an error-correcting code of perfect tensors, and the operator size is computed using the number of single-site physical operators that realize the logical operator. In both cases, we observe linear growth.