By calculating correlation functions for the Lieb-Liniger model based on the algebraic Bethe ansatz method, we conduct a finite-size scaling analysis of the eigenstate thermalization hypothesis (ETH) which is considered to be a possible mechanism of
thermalization in isolated quantum systems. We find that the ETH in the weak sense holds in the thermodynamic limit even for an integrable system although it does not hold in the strong sense. Based on the result of the finite-size scaling analysis, we compare the contribution of the weak ETH to thermalization with that of yet another thermalization mechanism, the typicality, and show that the former gives only a logarithmic correction to the latter.
We derive an upper bound on the difference between the long-time average and the microcanonical ensemble average of observables in isolated quantum systems. We propose, numerically verify, and analytically support a new hypothesis, eigenstate randomi
zation hypothesis (ERH), which implies that in the energy eigenbasis the diagonal elements of observables fluctuate randomly. We show that ERH includes eigenstate thermalization hypothesis (ETH) and makes the aforementioned bound vanishingly small. Moreover, ERH is applicable to integrable systems for which ETH breaks down. We argue that the range of the validity of ERH determines that of the microcanonical description.
