  We study the charge conductivity of the one-dimensional repulsive Hubbard
model at finite temperature using the method of dynamical quantum typicality,
focusing at half filling. This numerical approach allows us to obtain current
autocorrelation functions from systems with as many as 18 sites, way beyond the
range of standard exact diagonalization. Our data clearly suggest that the
charge Drude weight vanishes with a power law as a function of system size. The
low-frequency dependence of the conductivity is consistent with a finite dc
value and thus with diffusion, despite large finite-size effects. Furthermore,
we consider the mass-imbalanced Hubbard model for which the charge Drude weight
decays exponentially with system size, as expected for a non-integrable model.
We analyze the conductivity and diffusion constant as a function of the mass
imbalance and we observe that the conductivity of the lighter component
decreases exponentially fast with the mass-imbalance ratio. While in the
extreme limit of immobile heavy particles, the Falicov-Kimball model, there is
an effective Anderson-localization mechanism leading to a vanishing
conductivity of the lighter species, we resolve finite conductivities for an
inverse mass ratio of $\eta \gtrsim 0.25$.
