Localization in an interacting system with two degrees of freedom per site: Is charge disorder alone sufficient to localize the Hubbard model or is spin disorder also necessary?

Many-body localization, the inability of some isolated quantum systems with both interactions and disorder to achieve thermal equilibrium, has been studied primarily in systems with one degree of freedom per site. The addition of a second degree of freedom creates a new avenue for exploration and connects to the Hubbard model, a launch point for the study of a wide range of systems of current interest including transition metal oxides and cold atoms in optical lattices. With both charge and spin degrees of freedom at each site, is disorder in just one of these channels sufficient to cause the full system to be localized? Alternatively, does the degree of freedom without disorder delocalize the one which sees disorder? To what extent is the level of localization in one channel related to the disorder in the other channel? We consider the Hubbard model with both charge and spin disorder. We employ several measures of localization designed to treat spin and charge on equal footing. We find measures based on the local integrals of motion are consistent with those based on the dynamics. For sufficient disorder in one degree of freedom, only a small amount of disorder in the other degree of freedom localizes both degrees of freedom.