Band theory Hubbard model

Likewise the Hubbard model the periodic Anderson model (PAM) is a basic model in the theory of strongly correlated electron systems. It is destined for the description of the transition metals, lanthanides, actinides and their compositions including the heavy-fermion compounds. The model consists of two groups of electrons itinerant and localized ones (s and d electrons), the hybridization between them is admitted. The model is described by the following parameters the width of the s-electron band W, the energy of the atomic level e, the on-site Coulomb repulsion U of d-electrons with opposite spins, the parameter V of the... 

To cover the gap between them the Hubbard model Hamiltonian was quite generally accepted. This Hamiltonian apparently has the ability of mimicking the whole spectrum, from the free quasi-particle domain, at U=0, to the strongly correlated one, at U —> oo, where, for half-filled band systems, it renormalizes to the Heisenberg Hamiltonian, via Degenerate Perturbation Theory. Thence, the Heisenberg Hamiltonian was assumed to be acceptable only for rather small t/U values. 

The distance between two electrons at a given site is given as ri2. The electron wave function for one of the electrons is given as (p(ri) and the wave function for the second electron, with antiparallel spin, is Hubbard intra-atomic energy and it is not accounted for in conventional band theory, in which the independent electron approximation is invoked. Finally, it should also be noted that the Coulomb repulsion interaction had been introduced earlier in the Anderson model describing a magnetic impurity coupled to a conduction band (Anderson, 1961). In fact, it has been shown that the Hubbard Hamiltonian reduces to the Anderson model in the limit of infinite-dimensional (Hilbert) space (Izyumov, 1995). Hence, Eq. 7.3 is sometimes referred to as the Anderson-Hubbard repulsion term. 

Like in molecular quantum chemistry, the localized-delocalized antagonism is omnipresent in the theoretical literature on itinerant magnetism. On the one hand, the Hubbard model [292] and related theories for strongly correlated systems have been employed to study rare-earth and also transition metals. Since the latter do not have flat bands, extensions to the Hubbard theory are required [293-295] also, to make the model Hamiltonians (almost) exactly solvable, simplifications are introduced. On the other hand, density-functional theory is able to extract Stoner s parameters [296,297] for a self-consistent description of itinerant magnetism [298]. As has been illustrated before, the theoretical limits of the LDA became apparent from Fe phase stability problems (see Section 2.12.1) and were solved by using gradient corrections. The present status of DFT in the treatment of cooperative magnetism has also been reviewed [299]. 

Conjngated polymers differ from crystalline semiconductors and metals in several aspects and are often treated theoretically as a one-dimensional system. The formation of the band gap is explained taking into account either electron-phonon interactions or electron-electron interactions among 7t-electrons. If electron-phonon interaction dominates in real 7t-conjugated polymers, these systems could be treated using Peierls theory. In contrast, when electron-electron interactions dominate, the Hubbard model could be used to explain the physical properties of polymers. 

For the sake of clarity we shall first discuss in Section 6.2.1 ground-state properties in the framework of the unrestricted Hartree-Fock approximation. In Section 6.2.2 the theory is extended to finite temperatures by using a functional integral formalism including spin fluctuations. Finally, in Section 6.2.3 we analyze the problem of electron correlations by exact diagonalization of the simpler single-band Hubbard model. 

Katsufuji et al. used their results to estimate the critical value of the bandwidth-controlled Mott-Hubbard transition, (U/W)c 0.97, in the Ai-xCaxTiOs system. This value is below that estimated from quantum Monte Carlo simulations of the half-filled single band Hubbard model [72], (U/W)c 1.5 however a proper comparison between theory and experiment requires more careful consideration of both orbital degeneracy [149], which increases the critical value (U/W)c, and the deviation of the system from halffilling [150]. It is also worth noting for comparison that Ahn et al. estimate a value (U/W)c 2.7 from their measurements of the (Ca,Sr)Ru03 system [73]. 

Despite the qualitative agreement, there appear to be discrepancies and additional structures that do not seem to be attributable to contaminants or surface shifts. These anomalous features become more prominent for the systems that are expected to have the narrowest f bands and are most extreme for the heavy fermion class of uranium systems. In view of the large value of the imderlying non-interacting f band width A, it therefore seems reasonable to assume that the effect of Coulomb interactions may be introduced via perturbation theory. Thus, within the formalism of the Anderson impurity model, or even the Hubbard model, the spectra bear resemblance to a Lorentzian band of width A 2 e ( and since the 14-fold degenerate f band is expected to contain only 2 or 3 electrons, most of the spectral wei t is located above the Fermi level fi. 

Soos, Z. G. Ramasesha, S. Galvao, D. S. Band to correlated crossover in alternating Hubbard and Pariser Parr Pople chains nature of the lowest singlet excitation of conjugated polymers, Phys. Rev. Lett. 1985, 71, 1609-1612 Soos, Z. G. Ramasesha, S. Valence bond theory of linear Hubbard and Pariser-Parr-Pople models, Phys. Rev. 1984, B29, 5410-5422. 

---