Bounds on eigenstate thermalization

The eigenstate thermalization hypothesis (ETH), which asserts that every eigenstate of a many-body quantum system is indistinguishable from a thermal ensemble, plays a pivotal role in understanding thermalization of isolated quantum systems. Yet, no evidence has been obtained as to whether the ETH holds for all few-body operators in a chaotic system; such few-body operators include key quantities in statistical mechanics, such as the total magnetization, the momentum distributions, and their low-order thermal and quantum fluctuations. Here, we formulate a conjecture that for a generic nonintegrable system the ETH holds for all $m$-body operators with $m < {\alpha}_{\ast} N$ in the thermodynamic limit for some nonzero constant ${\alpha}_{\ast} > 0$. We first prove this statement for systems with Haar-distributed energy eigenstates to analytically motivate our conjecture. We then verify the conjecture for generic spin, Bose, and Fermi systems with local and few-body interactions by large-scale numerical calculations. Our results imply that generic systems satisfy the ETH for all few-body operators, including their thermal and quantum fluctuations.