Eigenstate Thermalization, Random Matrix Theory, and Behemoths

The eigenstate thermalization hypothesis (ETH) is one of the cornerstones of contemporary quantum statistical mechanics. The extent to which ETH holds for nonlocal operators is an open question that we partially address in this Letter. We report on the construction of highly nonlocal operators, behemoths, that are building blocks for various kinds of local and nonlocal operators. The behemoths have a singular distribution and width w similar to D-1 (D being the Hilbert space dimension). From there, one may construct local operators with the ordinary Gaussian distribution and w similar to D-1/2 in agreement with ETH. Extrapolation to even larger widths predicts sub-ETH behavior of typical nonlocal operators with w similar to D-delta, 0 < delta < 1/2. This operator construction is based on a deep analogy with random matrix theory and shows striking agreement with numerical simulations of nonintegrable many-body systems