Local Integrals of Motion in the Two-Site Anderson-Hubbard Model

It has been proposed that the states of fully many-body localized systems can be described in terms of conserved local pseudospins. While the states of any system can be expressed in terms of integrals of motion, the question of interest is whether these integrals of motion are local and if so on what length scale. The explicit identification of the optimally local pseudospins in specic systems is non-trivial. We consider the disordered Hubbard model and by studying a small system provide concrete examples of the form of integrals of motion. We explore ways of identifying the most local choice with charge disorder alone and with both spin and charge disorder.