Models Hubbard model

These findings are also supported by the results of a simple, exactly solvable model (Hubbard model for H2) where the ground state is correlated while the singly excited states are not [21]. For this model, the RPA yields bad results collapsing to complex eigenvalues for certain choices of the model parameters while the FOSEP approximation even gives the exact excitation energies. 

Because all the ab-initio methods mentioned above are quite time consuming, semi-empirical methods have been developed which replace certain integrals appearing in the Hartree Fock equations by parameters (or in some cases, functions) chosen so that the results fit the spectral properties of certain model compounds. Some of these methods are the Huckel model, Hubbard model, Pariser-Parr-Pople (PPP), CNDO, MNDO, (7), Since the first three of these methods are related and have played a significant role inthe work of the last few years, I will discuss them in more detail. 

Monte Carlo simulations [17, 18], the valence bond approach [19, 20], and g-ology [21-24] indicate that the Peierls instability in half-filled chains survives the presence of electron-electron interactions (at least, for some range of interaction parameters). This holds for a variety of different models, such as the Peierls-Hubbard model with the onsite Coulomb repulsion, or the Pariser-Parr-Pople model, where also long-range Coulomb interactions are taken into account ]2]. As the dimerization persists in the presence of electron-electron interactions, also the soliton concept survives. An important difference with the SSH model is that neu-... 

A great deal more could be said about models - to understand behavior like strong correlation, Coulomb blockade, and actual line shapes, it is necessary to use a number of empirical parameters, and a quite sophisticated form of density functional theory that deals with both static and dynamic correlation at a high level. Often this can be done only within a very simple representation of the electrons - something like the Hubbard model [51-53], which is very common in this situation. 

The spin distribution in the lower homologue [15 w = 2] was most elegantly studied by Takui et al. (1989) by means of a combination of ENDOR experiments and theoretical calculations within the framework of a generalized UHF Hubbard model (Teki et al., 1987a) and a Heisenberg model (Teki et al., 1987b). The spin distribution obtained is just as expected qualitatively in Fig. 7. 

Many electron systems such as molecules and quantum dots show the complex phenomena of electron correlation caused by Coulomb interactions. These phenomena can be described to some extent by the Hubbard model [76]. This is a simple model that captures the main physics of the problem and admits an exact solution in some special cases [77]. To calculate the entanglement for electrons described by this model, we will use Zanardi s measure, which is given in Fock space as the von Neumann entropy [78]. 

The Hamiltonian of the two-electron two-site Hubbard model can be written [77]... 

In the Hubbard model, the electron occupation of each site has four possibilities there are four possible local states at each site, v). = 0) -, I Ti) -... 

For the one-dimensional Hubbard model with half-filling electrons, we have iij) = (ny) =, M+ = j — w, and the entanglement is given by... 

Figure 4. Two-site Hubbard model. Upper curve is the entanglement calculated by the von Newmann entropy. The curves 5 1 and 5 2 are the correlation entropies of the exact wavefunction as defined in the text. The dashed line is the 5 2 for the combined wavefunction based on the range of V values. S for the combined wavefunction is zero.

In the Hubbard model, the electron occupation of each site has four possibilities there are four possible local states at each site, v). = 0)y, t) -, i) -, Ti)y The dimensions of the Hilbert space of an L-site system is 4 and IV] V2 Vf,) = vj)j can be used as basis vectors for the system. The entanglement of the jth site with the other sites is given in the previous section by Eq. (65). 

Consider the particle-hole symmetry of the one-dimensional Hubbard model. One can obtain w —U) = — w U), so the entanglement is an even function of U, Ej —U) = Ej U). The minimum of the entanglement is 1 as 1/ oo. As U +00, all the sites are singly occupied the only difference is the spin of the electrons on each site, which can be referred to as spin entanglement. As U —oo, all the sites are either doubly occupied or empty, which is referred... 

In this chapter, we will focus on the entanglement behavior in QPT for the two-dimensional array of quantum dots, which provide a suitable arena for implementation of quantum computation [88, 89, 103]. For this purpose, the real-space renormalization group technique [91] will be utilized and developed for the finite-size analysis of entanglement. The model that we will be using is the Hubbard model [83],... 

F. H. L. Essler, H. Frahm, F. Gohmann, A. Klumper, and V. E. Korepin, The One-Dimensional Hubbard Model, Cambridge University Press, Cambridge, 2005. 

In order to take into account these intra-atomic terms, and in a way similar to the Stoner s model, Hubbard ), see also adds to the Hamiltonian (11) an exchange interaction term ... 

We have discussed, in Chap. A, the Hubbard model for localization vs. itineracy in narrow bands. In this model, it was shown, for a simple case ( hydrogen case) involving one uncoupled electron in a shell, that a splitting of the narrow band in two sub-bands occurs when the Hubbard condition (Uh = W) is not satisfied. The two sub-bands describe two situations ... 

Equation (20) may be seen as the combination of the two processes of direct and inverse photoemission, when the 5 f shell retains a strong character of localization (in case of itinerant 5fs, the Hubbard model predicts that the kinetic energy due to itineracy creates statistical fluctuations between the polar states, so that the itinerant character is lost). 

Within the actinide series Pu is the most intriguing element. On the basis of the Hubbard model, and taking into account an unhybridized bandwidth Wf (due only to f-f overlapping), the Un/Wf ratio is 0.7 for U and 3 for Pu in fact, one would have expected already for Pu a 5 f electron localization, since Uh > Wf. However, a hybridization of 5 f with (6d7s) states broadens the 5f bandwidth and delays the Mott-like transition (see Chap. A) from Pu to Am This influences many properties of Pu metaf . ... 

Among numerous theoretical approaches, the Gutzwiller method [11,12] provides a transparent physical interpretation in term of atomic configurations of a given site. Originally it was applied to the one-band Hubbard model Hamiltonian [13] ... 

Theoretical interpretation is incomplete early measurements on La-Sr-Cu-O by Hundley et al. (76) were suggested to indicate a phonon drag to explain some features although more recently a narrow band Hubbard model has been developed by Fisher (77). 

Electron correlation plays an important role in determining the electronic structures of many solids. Hubbard (1963) treated the correlation problem in terms of the parameter, U. Figure 6.2 shows how U varies with the band-width W, resulting in the overlap of the upper and lower Hubbard states (or in the disappearance of the band gap). In NiO, there is a splitting between the upper and lower Hubbard bands since IV relative values of U and W determine the electronic structure of transition-metal compounds. Unfortunately, it is difficult to obtain reliable values of U. The Hubbard model takes into account only the d orbitals of the transition metal (single band model). One has to include the mixing of the oxygen p and metal d orbitals in a more realistic treatment. It would also be necessary to take into account the presence of mixed-valence of a metal (e.g. Cu ", Cu ). 

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