  While many-body localization has primarily been studied in systems with a
single local degree of freedom, experimental studies of many-body localization
in cold atom systems motivate exploration of the disordered Hubbard model. With
two coupled local degrees of freedom it is natural to ask how localization in
spin relates to disorder in charge and vice versa. Most prior work has
addressed disorder in only one of these sectors and often has not used measures
of localization which distinguish between charge and spin. Here we explore
localization in the Hubbard model with a wide range of independent values of
charge and spin disorder, using measures of localization based on charge and
spin-specific integrals of motion. Our results demonstrate symmetry between the
response of the spin to charge disorder and vice versa, and we find very weak
disorder in one channel, so long as the disorder in the other channel is
sufficiently strong, results in localization in both channels. The strength of
disorder required in the less disordered channel declines as the system size
increases. Further, the weaker the disorder in the less-disordered channel, the
longer the time scale at which localization appears in the dynamics of this
degree of freedom.
