# Electronic correlations in narrow-band solids

## Abstract

The Hubbard Hamiltonian is rederived from the full many-body Hamiltonian with the assumption that only intrasite correlations are important. It is shown to be exact in both the Hartree-Fock and narrow-band limits, provided the appropriate linearization procedure is adopted in the former case. Real-time and imaginary-time Green's functions are derived for the cases intermediate between the Hartree-Fock and narrow-band limits, and a long-standing puzzle with regard to the number of electrons in the upper pseudoparticle band is cleared up. The chemical potential and total energy of the system are calculated in the narrow-band limit and are shown to be identical with those derived from an effective one-electron representation. It is shown that because these quantities depend only on the number of doubly occupied sites, important transport parameters such as electrical conductivity and thermoelectric power can be calculated from the effective one-electron representation, without the necessity of evaluating the two-particle Green's function. For finite bands, the total energy in the Hubbard model is shown to give the exact result, in contradiction to a previous calculation. It is shown that thermodynamic quantities such as the total energy and chemical potential which depend only on derivatives of the grand partition function are independent of the presence or absence of magnetic ordering, but that the entropy is not. Thus a study of the insulatormetal phase transitions is very sensitive to magnetic ordering. © 1979 The American Physical Society.