Convergence of eigenstate expectation values with system size

TL;DR

This paper investigates how quickly eigenstate expectation values in quantum lattice systems approach a smooth function as system size increases, establishing a universal lower bound on the convergence rate.

Contribution

It proves a universal lower bound of 1/O(N) for the convergence rate of eigenstate expectation values in translation-invariant quantum systems, regardless of integrability.

Findings

Lower bound of 1/O(N) on convergence rate for most local operators.

Bound holds regardless of system integrability or chaos.

Saturation of the bound in systems satisfying eigenstate thermalization hypothesis.

Abstract

Understanding the asymptotic behavior of physical quantities in the thermodynamic limit is a fundamental problem in statistical mechanics. In this paper, we study how fast the eigenstate expectation values of a local operator converge to a smooth function of energy density as the system size diverges. In translation-invariant quantum lattice systems in any spatial dimension, we prove that for all but a measure zero set of local operators, the deviations of finite-size eigenstate expectation values from the aforementioned smooth function are lower bounded by $1/O(N)$, where $N$ is the system size. The lower bound holds regardless of the integrability or chaoticity of the model, and is saturated in systems satisfying the eigenstate thermalization hypothesis.