Assuming that, in the narrow-band region, the exact Green's function of the Hubbard model essentially retains the two-peak structure proposed by Hubbard, we derive its consequences as rigorously as possible. When a quasielectron is added to the ground state of
N electrons in the narrow-band region of the Hubbard model, the whole system is shown to relax due to the strong interaction
I, thus yielding a relaxation energy $\ensuremath{\Delta}E$ between the quasielectron and the original
N electrons. This explains the conclusion in the preceding paper that a gap $\ensuremath{\Delta}E$ analogous to the energy gap in superconductivity appears between the quasielectron spectrum and the chemical potential for any occupation
n regardless of whether the lower band is exactly filled or not and that the system cannot be a normal metal. We further found that the ground state is, in fact, a bound state, and electrons involved do not obey the Landau theory of quasiparticles. However, the electron Green's function describes the motion of an electron added to this
N-electron ground state without the relaxation of the original
N electrons, and hence this added electron behaves exactly like a Landau quasiparticle and can be represented by the quasiparticle operator ${\mathrm{A}}_{k\ensuremath{\sigma}}^{\ifmmode\dagger\else\textdagger\fi{}}$ introduced by Luttinger and Nozi\`eres. Only if the original
N electrons are allowed to relax is the energy of the system reduced by $\ensuremath{\Delta}E$, yielding the energy gap $\ensuremath{\Delta}E$. When a quasielectron is removed from the system of
N+1 electrons, the same relaxation energy $\ensuremath{\Delta}E$ is removed, and again the gap $\ensuremath{\Delta}E$ is introduced between the chemical potential $\ensuremath{\mu}$ and the quasihole spectrum, thus creating a gap $2\ensuremath{\Delta}E$ between the quasielectron and quasihole spectra for any occupation
n. The Fermi level, lying at the center of the gap, never intersects the quasiparticle spectra and therefore, Luttinger's theorem that the volume within the Fermi surface is independent of interactions and equal to the value in the noninteracting limit does not apply to the Hubbard model in the narrow-band region. However, this conclusion does not immediately imply that the system is an insulator. Instead, it is shown that a current-carrying state analogous to the condensate state found by Fr\"ohlich appears under certain conditions. When the number of electrons
N is nearly or exactly equal to the number of atoms
Na, quasiparticlelike behavior disappears completely, making the system insulating. The only exception is the pathological limit $I=\ensuremath{\infty}$, where a metallic state with a Fermi volume twice that in the noninteracting limit is found.
The self-consistent theory of the correlation effects in Highly Correlated Systems(HCS) is presented. The novel Irreducible Green's Functions(IGF) method is discused in detail for the Hubbard model and random Hubbard model. The interpolative solution for the quasiparticle spectrum, which is valid for both the atomic and band limit is obtained. The (IGF) method permits to calculate the quasiparticle spectra of many-particle systems with the complicated spectra and strong interaction in a very natural and compact way. The inelastic scattering corrections leads to the damping of the quasiparticles and are the main topic of the present consideration. The calculation of the damping has been done in a self-consistent way for both limits. For the random Hubbard model the weak coupling case has been considered and the self-energy operator has been calculated using the combination of the IGF method and Coherent Potential Approximation (CPA). The other applications of the method to s-f model, Anderson model, Heisenberg antiferromagnet, electron-phonon interaction models are discussed briefly.
In the general frame of a modified many-body perturbation theory, the single-band Hubbard model is treated in the limit of weak electron-electron interaction. In this modification the decomposition of the Hamiltonian into an effective free system and an effective perturbation is rearranged iteratively in each order of the formalism, resulting in a successive reduction of the interaction part. In comparison with conventional perturbation techniques, this means an extended range of validity. Imposing thermodynamic self-consistency, we calculate the wave-vector-dependent electronic self-energy up to second order and analyze the influence of the Coulomb correlations. Several well-known physical properties of the Hubbard model are recovered in characteristic single-particle quantities. Therefore, the effect of the band occupation n and the electron-electron repulsion U on spectral densities, band structures, densities of states, magnetization, etc. is investigated in detail. Some of the most remarkable results are: first, a splitting of the free Bloch band into two quasiparticle subbands when the coupling strength reaches moderate values. These subbands are linked by a small damping-induced background in the density of states and are separated roughly by an energy of the order of U. In addition, the upper subband is shifted to higher energies in the spectrum proportionally to U while the lower one is nearly fixed. Second, we found a paramagnetic-ferromagnetic phase diagram that has a critical coupling strength
Uc and exhibits strong indications of a lower critical occupation number
nc. There is no spontaneous magnetization in the system for parameters smaller than these values. Compared with the Stoner model, one thus obtains striking improvements. In many cases our conclusions for the weak-coupling limit are qualitatively in good agreement with well-known results in the strong-correlation regime.
Some correlation effects are studied in a narrow-band model with electron–hole asymmetry. A peculiar feature of the model is that it takes into account the hopping integral associated with electron–electron interaction (correlated hopping). The method of Green’s functions is used to find the quasiparticle energy spectrum, as well as expressions for the upper and lower Hubbard subband widths. It is shown that the specific features of the model lead in a number of cases to consequences differing significantly from those following from the Hubbard model. The obtained results mainly illustrate the peculiarities of the model in the low-temperature region and are used for interpreting the physical properties of systems with narrow energy bands.
An analytic approach is presented of electronic one-particle spectra of the one-band Hubbard model at half filling in the antiferromagnetic phase. Starting from the strong-coupling regime U\ensuremath{\gg}:t, a projection technique is used to set up self-consistent coupled equations for the electron Green function, which are valid down to values U\ensuremath{\approx}t. The self-consistent equation for the hole propagator is a direct generalization of the one found from the t-J model. This gives further support to the ``string'' picture, where propagation of holes creates strings of overturned spins with which the holes interact. In the present work hopping of holes (or electrons) with up spin on the down sublattice is also taken into acount, as well as transitions between the lower and upper Hubbard bands. These are shown to change significantly the incoherent part of the t-J model spectra, by smearing out the shake-off peaks, reminiscent of higher bound string states due to multispin scattering. Coherent (quasiparticle) peaks exist at the band edges, on both sides of the insulating gap. We show that with decreasing U the quasiparticle concept loses its meaning for wave vectors at the center of the magnetic Brillouin zone (MBZ). For large values of U the dispersion of the quasiparticle is found to scale with its band width, which is of order J. Extrema are always found at k=(\ensuremath{\pi}/2,\ensuremath{\pi}/2). The weight of the quasiparticle at this k value decreases logarithmically with increasing U. In the strong-coupling limit the spectrum tends to be symmetric, i.e., to become an even function of the frequency around the chemical potential, for any wave vector. For small values of U the dispersion at the edge of the MBZ flattens away, as expected when approaching the noninteracting limit. The spectral function in this regime, for wave vectors away from the edge of the MBZ, is concentrated mainly on one side of the chemical potential.