Single-particle excitations under coexisting electron correlation and disorder: A numerical study of the anderson-hubbard model

Interplay of electron correlation and randomness is studied using the Anderson-Hubbard model within the Hartree-Fock (HF) approximation. Under the coexistence of short-range interaction and diagonal disorder, we obtain the ground-state phase diagram in three dimensions (3D), 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 1D or even faster in 2D and 3D toward the Fermi energy. We call such a gap a soft Hubbard gap. Moreover, exact-diagonalization results for 1D support the formation of a soft Hubbard gap beyond the mean-field level. The formation of the soft Hubbard gap cannot be attributed to the conventional theory by Efros and Shklovskii (ES) owing the emergence of soft gaps to the long-range Coulomb interaction. Indeed, on the basis of a multivalley energy landscape, we propose a phenomenological scaling theory, which predicts a scaling of the DOS, A in energy E as A(E) α exp[γlog |E- E_j|^d. Here, d is the spatial dimension, E_F is the Fermi energy, and γ is a non universal constant. This scaling is 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 ES scaling. 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 for SrRu_1-xTi _xO₃.