Operator Size Distribution in Large $N$ Quantum Mechanics of Majorana Fermions

TL;DR

This paper derives the complete time evolution of operator size distribution in large N Majorana fermion quantum mechanics, providing insights into operator growth and chaos in these systems.

Contribution

It introduces a formalism to compute the full time evolution of operator size distribution in large N Majorana fermion models, including SYK variants.

Findings

Derived the full time evolution of operator size distribution.

Applied the formalism to Brownian SYK and large q SYK models.

Provided insights into operator growth and quantum chaos.

Abstract

Under the Heisenberg evolution in chaotic quantum systems, initially simple operators evolve into complicated ones and ultimately cover the whole operator space. We study the growth of the operator ``size'' in this process, which is related to the out-of-time-order correlator (OTOC). We derive the full time evolution of the size distribution in large $N$ quantum mechanics of Majorana fermions. As examples, we apply the formalism to the Brownian SYK model (infinite temperature) and the large $q$ SYK model (finite temperature).