Stability of the split-band solution and energy gap in the narrow-band region of the Hubbard model

By inserting quasielectron energies $\omega$ calculated from the fully renormalized Green's function of the Hubbard model obtained in the preceding paper into the exact expression of Galitskii and Migdal, the ground-state energy, the chemical potential, and the dynamic- and thermodynamic-stability conditions are calculated in the narrow-band region. The results show that as long as the interaction energy $I$ is finite, electrons in the narrow-band region do not obey the Landau theory of Fermi liquids, and a gap appears between the lowest quasielectron energy $\omega$ and the chemical potential $\mu$ for any occupation $n$, regardless of whether the lower band is exactly filled or not. This unusual behavior is possible because, when an electron is added to the system of $N$ electrons, the whole system relaxes due to the strong interaction, introducing a relaxation energy difference between the two quantities. We also show that all previous solutions which exhibit the split-band structure, including Hubbard's work, yield the same conclusion that electrons do not behave like Landau quasiparticles. However, the energy gap is calculated to be negative at least for some occupations $n$, demonstrating the dynamic instability of those solutions. They also exhibit thermodynamic instability for certain occupations, while the fully renormalized solution, having sufficient electron correlations built in, satisfies the dynamic and thermodynamic stability conditions for all occupations. When the lower band is nearly filled, the nature of the solution is shown to change, making the coherent motion of electrons with fixed $k$ values more difficult. In the pathological limit where $I=\infty$, however, the gap vanishes, yielding a metallic state.