Within the one-dimensional Hubbard model linear closed chains with various numbers of sites are considered in the self-consistent random phase approximation (SCRPA). Excellent results with a minimal numerical effort are obtained for
(2+4n)-site cases, confirming earlier results with this theory for other models. However, the
4n-site cases need further consideration. The SCRPA solves the two-site problem exactly. It therefore contains the two-electron and high-density Fermi gas limits correctly.
We are interested in the possibility that a generalization of Self Consistent Random Phase Approximation (SCRPA) to the Extended Hubbard Model can lead us to accurate estimations of the ground state energy, for closed chains in one dimension with periodic boundary conditions: N=2 (two states problem). We have considered an extended Hubbard model including on-site and off-site interactions, we have compared the SCRPA with a Direct Analytical(DA) results for the ground state energy, and have shown that SCRPA solves the two states problem exactly for any value of U (on-site interaction energy), V2 (off-site interaction energy with opposite spins) and (off-site interaction energy with same spins).
The self consistent random phase approximation (SCRPA) and a direct analytical (DA) method are proposed to solve the Extended Hubbard Model (EHM) in one dimension (1D). We have considered an EHM including on-site and off-site interactions for closed chains in 1D with periodic boundary conditions. The comparison of the SCRPA results with the ones obtained by a DA approach shows that the SCRPA treats the problem of these closed chains in a rigorous manner. The analysis of the nearest–neighbour repulsion effect on the dynamics of our closed chains shows that this repulsive interaction between the electrons of the neighbouring atoms induces supplementary conductivity, since, the SCRPA energygap vanishes when these closed chains are governed by a strong repulsive on-site interaction and intermediate nearest–neighbour repulsion.
The atomic limit of the Hubbard model is a simple single-site problem which can be solved exactly, and all one- and two-particle Green's functions can be obtained analytically. These solutions can thus serve as a means of critiquing the success of various approximate theories which might be applied to the full Hubbard model. In particular, we have examined the
T-matrix approximation for the attractive Hubbard model in the atomic limit, which should give reasonable results at low electronic densities, if one can avoid the spurious phase transition that results when a fully non-self-consistent
T-matrix approximation is employed---previously we have shown that any level of self-consistency guarantees that this phase transition is correctly suppressed to zero temperature in two dimensions or less. Here, a minimally self-consistent
T-matrix approximation is shown to be successful in reproducing the exact results for the atomic limit, while fully self-consistent
T-matrix results do not agree with the known solutions. Of particular note is that the minimally self-consistent
T-matrix approximation reproduces not only one- and two-particle (static) thermodynamic quantities, but it also exactly reproduces the one-particle spectral function at low but nonzero temperatures. We also make a comparison to the two-particle self-consistent approach of Vilk and Tremblay, and find that the minimally self-consistent
T-matrix theory can give better results over a broader temperature range.
Using the extended Hubbard model with the Self Consistent Random Phase Approximation (SCRPA), we study the intersite interaction effect on the dynamics of N electrons. We consider the extended Hubbard model containing intrasite and intersite interactions, and we apply this model to a system of two interacting atoms, where each atom presents one free electron. The resolution of this tow sites problem is done by the SCRPA approximation; our calculations allow as studying the intersite interaction effect on the excitation energies of system. We have showed that the attractive interaction between the electrons of the neighbour’s atoms is the origin of supplementary conductivity of the system.
The diagrammatic formulation of the single-site approximation to the Hubbard model immediately yields the exact solution to the Falikov-Kimball model in infinite dimensions. By an extension of the alloy analogy, we then derive a two-fluid model, dividing self-consistently the electron system among itinerating and relatively localized particles. Our theory coincides with the CPA only the at half-filling and u=∞. We start by studying the mean field approximation to our theory and identify a phase boundary in the particle density-interaction strength (n−u) plane at which the Fermi liquid model breaks down. Then the effect of fluctuations in the Fermi liquid regime is considered and, finally, we construct, at half-filling, a model which interpolates between the Fermi liquid and the Mott insulating phases.