Identifying the closeness of eigenstates in quantum many-body systems**Project supported by the Natural Science Foundation of Zhejiang Province, China (Grant No. LY16A050004), the Fundamental Research Funds for the Central Universities, China (Grant No. 2017FZA3005), and the National Natural Science Foundation of China (Grant No. 11475146).

Abstract

We propose a quantity called modulus fidelity to measure the closeness of two quantum pure states. We use it to investigate the closeness of eigenstates in one-dimensional hard-core bosons. When the system is integrable, eigenstates close to their neighbor or not, which leads to a large fluctuation in the distribution of modulus fidelity. When the system becomes chaos, the fluctuation is reduced dramatically, which indicates all eigenstates become close to each other. It is also found that two kind of closeness, i.e., closeness of eigenstates and closeness of eigenvalues, are not correlated at integrability but correlated at chaos. We also propose that the closeness of eigenstates is the underlying mechanism of eigenstate thermalization hypothesis (ETH) which explains the thermalization in quantum many-body systems.