  Interacting two-component Fermi gases loaded in a one-dimensional (1D)
lattice and subjected to an harmonic trapping potential exhibit interesting
compound phases in which fluid regions coexist with local Mott-insulator and/or
band-insulator regions. Motivated by experiments on cold atoms inside
disordered optical lattices, we present a theoretical study of the effects of a
correlated random potential on these ground-state phases. We employ a lattice
version of density-functional theory within the local-density approximation to
determine the density distribution of fermions in these phases. The
exchange-correlation potential is obtained from the Lieb-Wu exact solution of
Fermi-Hubbard model. On-site disorder (with and without Gaussian correlations)
and harmonic trap are treated as external potentials. We find that disorder has
two main effects: (i) it destroys the local insulating regions if it is
sufficiently strong compared with the on-site atom-atom repulsion, and (ii) it
induces an anomaly in the inverse compressibility at low density from quenching
of percolation. For sufficiently large disorder correlation length the
enhancement in the inverse compressibility diminishes.
