Hamiltonian extended Hubbard

We adopt an extended Hubbard Hamiltonian postulated for the active space of the form... 

The properties of low-dimensional organic conductors are determined by different interactions and instabilities (see, e.g., Chapter 2). It is commonly assumed that it can be treated as in ordinary metals with a frozen long-range interaction part, while the short-range part gives rise to quasiparticles with screened interactions. Thus the proper model Hamiltonian is of the extended Hubbard type [11]. 

Most organic conductors behave as quasi-one-dimensional systems, at least at high temperatures. It is generally accepted that due to the narrow band-widths, the strongest interactions are the electron-electron Coulomb interaction, U for two electrons on the same site, and V for electrons in nearest-neighbor sites, in agreement with the extended Hubbard Hamiltonian, which is usually taken as a good approximation. 

To see the relevance of the above to cuprate superconductors consider an effective electronic Hamiltonian as the sum of one-body and two-body effective interactions. We adopt an extended Hubbard Hamiltonian postulated for the active space of the form... 

The theory evolved from an examination of ab initio configuration-interaction (Cl) wave functions for a number of small clusters, as well as from a study of a semi-empirical two-band extended Hubbard Hamiltonian which incorporates the full periodicity of the lattice. The parameters wUch diaracterize the latter were chosen to be consistent with the ab initio results. For La2Cu04, both approaches predict the experimentally observed anti-ferromagnetic spin interaction when the now froniliar Cu dx2.y2... 

In the physics literature, the model invented to go beyond the severely truncated electron-repulsion range of the Hubbard model has been the extended Hubbard Hamiltonian or the U-V model. Here, the intersite interactions are introduced between bonded sites through another independent phenomenological interaction parameter V [26] from which the model derives its name. The Hamiltonian is given... 

A quantitative approach is to define these Coulomb interactions, not only the on-site interaction but also several near-neighbor interactions, which cannot be negligible, conq)ared to the bandwidth. A valuable approximation is to consider the two first terms. TTie extended Hubbard hamiltonian [20] is, in second quantization notation (site representation) ... 

One should also mention that a more realistic modeling of a quarter-filled system would require the inclusion of at least the nearest-neighbor Coulomb repulsion terms in addition to the on-site ones. That means switching from the original Hubbard to the extended Hubbard model Hamiltonian ... 

An intriguing possibility is offered by the interaction engineering discussed above in the context of the realization of effective lattice models where particles interact via exotic (extended) Hubbard Hamiltonians. An example of this is given in Ref. [85], where it is shown how to engineer the following Hubbard-like Hamiltonian... 

Hiickel and PPP theory emphasize common aspects of 7T electrons in molecules whose geometry is specified in advance. The backbones of the representative conjugated polymers in Fig. 6.2, for example, immediately fix the Hiickel or PPP Hamiltonians. Effects due to substituents R or sulfur heteroatoms in polythiophene (PT) are approximated by site energies or neglected as perturbations of TT-TT tran.sitions. Such a priori information about Ha and V(/ ) comes from molecules and contrasts sharply with the adjustable parameters of extended Hubbard models. The a- conjugation encountered in polysi-... 

The vibrational consequences of 7r-electron fluctuations discussed in Sections II and III drew on both molecular spectroscopy and solid-state physics. The analysis of NLO and EA spectra in Section IV combined PPP models for molecules with quantum cell models of alternating chains. We proposed at the outset to relate the conjugated polymers in Fig. 6.2 to alternating Fliickel or PPP chains and have so far discussed vibrational and optical implications of 7r-electron models rather than the Hamiltonian, Eq. (7), or its mathematical properties. The analysis holds for any H(8) with appropriate vibrational or optical susceptibilities. Equation (7) is sufficiently general to encompass Hiickel, Hubbard, extended Hubbard, PPP, and other models with suitable choices of U and Vp,. This generality is an extremely useful feature of solid-state models. 

The two-dimensional t-J model was shown to describe correctly the low-energy part of the spectrum of the extended Hubbard model, which gives a realistic description of the Cu-0 plane of cuprates [46-48]. The Hamiltonian of the t-J model reads... 

The magnetic susceptibility measurements allow a determination of the Hubbard U value. From the magnetic data [489] has been estimated 17/4/ in 2-chain compounds such as TTF-TCNQ. This approach has been generalized in [490]. The authors of these papers used an extended Hubbard Hamiltonian and they showed that the magnetic susceptibility varies strongly and systematically as a function of band filling. An important result shows that the extended Hubbard data cannot be mapped systematically onto the results of a Hubbard model with an effective U value. 

In this Section the ferromagnetic and the antiferromagnetic Heisenberg exchange Hamiltonians are derived from the Hiickel-Hubbard Hamiltonian in which the value of the coupling constant, z takes on values from - 1 to + 1. The M = N = 2, linear, extended Hiickel-Hubbard spectrum is plotted in Fig. 4.1. (Compare with Fig. 3.1)... 

The Heisenberg Hamiltonians are derived by the application of second-order perturbation theory to the extended Huckel-Hubbard Hamiltonian at z close to -l(ferromagnet) and at z close to + 1 (antiferromagnet) respectively. 

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