Simple slow operators and quantum thermalization

TL;DR

This paper rigorously links the thermalization process of typical states to the dynamics of local operators through the concept of simple slow operators (SSOs), revealing that the existence or absence of SSOs determines thermalization behavior.

Contribution

It introduces the concept of simple slow operators and establishes a rigorous relation between their existence and the thermalization of typical initial states.

Findings

Existence of SSOs correlates with non-thermalization of initial states.

Absence of SSOs implies that typical states thermalize.

Operator size and ensemble variance norm are key to understanding operator growth and thermalization.

Abstract

We establish a rigorous relation between the thermalization of typical initial states and the dynamics of local operators. We introduce a concept of simple slow operators (SSOs), defined as operators that have a small commutator with the Hamiltonian and have significant small-sized components. We show that if typical initial states (drawn from a low-complexity state ensemble) do not thermalize on timescale $t$, then SSOs must exist that are approximately conserved up to timescale $t$. Equivalently, the absence of SSOs implies that typical initial states thermalize. We establish these results by introducing the concept of an ensemble variance norm of an operator, defined as the typical magnitude of the expectation value of that operator with respect to states in the ensemble. For low-entanglement ensembles, the norm is related to operator sizes, allowing us to establish a direct link between operator growth and thermalization.