Atypicality of Most Few-Body Observables

Understand

The eigenstate thermalization hypothesis (ETH), which dictates that all diagonal matrix elements within a small energy shell be almost equal, is a major candidate to explain thermalization in isolated quantum systems.

• According to the typicality argument, 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 typicality argument does not apply to most few-body observables for few-body Hamiltonians when the width of the energy shell decreases at most polynomially with increasing the size of the system.