  Interplay of electron correlation and randomness is studied by using the
Anderson-Hubbard model within the Hartree-Fock approximation. Under the
coexistence of short-range interaction and diagonal disorder, we obtain the
ground-state phase diagram in three dimensions, which includes an
antiferromagnetic insulator, an antiferromagnetic metal, a paramagnetic
insulator (Anderson-localized insulator) and a paramagnetic metal. Although
only the short-range interaction is present in this model, we find
unconventional soft gaps in the insulating phases irrespective of electron
filling, spatial dimensions and long-range order, where the single-particle
density of states (DOS) vanishes with a power-law scaling in one dimension (1D)
or even faster in two dimensions (2D) and three dimensions (3D) toward the
Fermi energy. We call it soft Hubbard gap. Moreover, exact-diagonalization
results in 1D support the formation of the soft Hubbard gap beyond the
mean-field level. The formation of the soft Hubbard gap cannot be attributed to
a conventional theory by Efros and Shklovskii (ES) owing the emergence of soft
gaps to the long-range Coulomb interaction. Indeed, based on a picture of
multivalley energy landscape, we propose a phenomenological scaling theory,
which predicts a scaling of the DOS in perfect agreement with the numerical
results. We further discuss a correction of the scaling of the DOS by the
long-range part of the Coulomb interaction, which modifies the scaling of Efros
and Shklovskii. Furthermore, explicit formulae for the temperature dependence
of the DC resistivity via variable-range hopping under the influence of the
soft gaps are derived. Finally, we compare the present theory with experimental
results of SrRu_{1-x}Ti_xO_3.
