On the origin of exponential operator growth in Hilbert space

TL;DR

This paper links exponential operator growth in quantum many-body systems to off-diagonal decay of operator matrix elements, revealing a universal bound on growth rate independent of chaos or interactions.

Contribution

It unifies operator dynamics and matrix element decay, providing a microscopic explanation for exponential operator growth in Hilbert space.

Findings

Exponential operator growth correlates with exponential off-diagonal decay.

Growth rate saturates the universal bound when decay is algebraic or slower.

Operator growth origin is independent of chaos, dimensionality, or many-body interactions.

Abstract

The question of thermalization in quantum many-body systems has long been studied through the properties of matrix elements of operators corresponding to local observables. More recently, the focus has shifted to the dynamics of operators, which lead to seminal works proposing universal bounds on the rate of operator growth. In this work, we unify these two approaches: we show that exponential operator growth in Hilbert space, as measured by Krylov complexity, is governed by an exponential off-diagonal decay of the operator matrix elements in the system eigenbasis. When this decay is algebraic or slower, the growth rate saturates the universal bound, thereby establishing a microscopic origin of operator growth which is independent of chaos, dimensionality or the presence of many-body interactions.