  Using the one-dimensional Fermi-Hubbard model we study the effects of lattice
imperfections on high-harmonic generation (HHG) from correlated systems. We
simulate such imperfections by randomly modifying the chemical potential across
the individual lattice sites. We control the degree of correlation by varying
the Hubbard $U$. In the limit of vanishing $U$, this approach results in
Anderson localization. For non-vanishing $U$, we explain the spectral
observations using a qualitative picture in which correlation and the
imperfections may balance each other out, causing Bloch-like transitions, i.e.,
transitions similar to those occurring in the case with small or vanishing $U$
and with vanishing imperfection-induced energy gaps, even though the dynamics
take place under conditions of high $U$ and severe imperfections. We verify
this picture by studying HHG spectra where imperfections are found to cause
gain in the HHG spectra. The spectral gain is mainly in high harmonic orders
for low correlation and low harmonic orders for high correlation. We explain
this using the following qualitative picture. For low correlation, the addition
of imperfections gives rise to significant energy gaps in the system,
corresponding to high harmonics, and when the correlation is massive the
addition of imperfections balances out the correlation, resulting in smaller
energy gaps, meaning lower harmonic orders.
  We further demonstrate how in the unbalanced cases, i.e., high $U$ and
relatively small amount of imperfection or vice versa, the dynamics are largely
adiabatic, and how in the balanced cases, with similar magnitude of correlation
and imperfection, the dynamics are qualitatively similar.
