Particle fluctuations and the failure of simple effective models for many-body localized phases

We investigate and compare the particle number fluctuations in the putative many-body localized (MBL) phase of a spinless fermion model with potential disorder and nearest-neighbor interactions with those in the non-interacting case (Anderson localization) and in effective models where only interaction terms diagonal in the Anderson basis are kept. We demonstrate that these types of simple effective models cannot account for the particle number fluctuations observed in the MBL phase of the microscopic model. This implies that assisted and pair hopping terms-generated when transforming the microscopic Hamiltonian into the Anderson basis-cannot be neglected even at strong disorder and weak interactions. As a consequence, it appears questionable if the microscopic model possesses an exponential number of exactly conserved local charges. If such a set of conserved local charges does not exist, then particles are expected to ultimately delocalize for any finite disorder strength.