Complexity, typicality, and eigenstate thermalization in fermion systems

The discovery of quantum many-body scar (QMBS) states has revealed the possibility of having states with low entanglement entropy (EE) that violate the eigenstate thermalization hypothesis (ETH) in nonintegrable systems. Such states with low EE are rare but naturally exist in the integrable system of free fermions. By representing the occupation pattern of each free fermion eigenstate as a classical binary string, we find that the Kolmogorov complexity of the string correctly captures the scaling behavior of EE for the eigenstate. This allows us to distinguish typical and atypical eigenstates directly by their intrinsic complexity. We further reveal that the fraction of atypical eigenstates which do not thermalize in the one-dimensional free fermion system vanishes exponentially in the thermodynamic limit. By introducing an arbitrarily weak two-body interaction between the fermions, we demonstrate analytically that those atypical states would be always eliminated. Specifically, we show that the probability of having a state with EE satisfying a sub-volume scaling law decreases double exponentially as the system size. Thus, our results provide a quantitative argument for the ETH and disappearance of QMBS states in weakly interacting fermion systems.