Atypicality of Most Few-Body Observables

The eigenstate thermalization hypothesis (ETH), which dictates that all diagonal matrix elements within a small energy shell be almost equal, is a promising candidate to explain thermalization in isolated quantum systems. According to the typicality argument over the Haar measure, the maximum variations of such matrix elements should decrease exponentially with increasing the size of the system, which implies the ETH. We show, however, that the argument does not apply to most few-body observables for few-body Hamiltonians unless the width of the energy shell decreases exponentially with increasing the size of the system.