The eigenstate distribution fluctuation from thermal to localized transitions

We study the thermalization of a quenched disordered Bose-Hubbard model. By considering the eigenstate distribution fluctuation, we show that the thermal to many-body localized transition is always connected to a minimum of this distribution fluctuation. We also observe a Mott-localized regime, where the system fails to thermalize due to the strong on-site repulsion. At last, we show how to detect this eigenstate distribution fluctuation in cold atom systems, which is equivalent to measure the Loschmidt echo of the system. Our work suggests a way to measure the thermal to localized transitions in experiments, especially for a large system.