Anderson-Hamiltonian localization

When the two-peaked structures were unambiguously observed for Ce and many of its compounds, several theoretical approaches were taken to explain their origin. Most of these studies of valence and 4f photoemission in Ce began with the assumption that the 4f state was best described as localized. Authors disagreed, however, not only on which terms were important in the Hamiltonian, but also on whether the double-peaked structure was present in the ground state or not. Many began with the Anderson Hamiltonian in which the 4f system is treated as an impurity on one site of the system (the origin) which can be written as... 

Allen and Martin (1982) and Lavagna et al. (1982, 1983) proposed that the y-a transition in Ce was related to the Kondo effect. Since Kondo systems have densities of states with two peaks, one very close to the Fermi level and one below, this offered a possible explanation of the two peaks in the photoelectron spectra, assuming that the photoabsorption process would transfer the structure in the density of states to the emitted spectrum. Detailed calculations of a photoelectron spectrum were not, however, carried out. The Kondo (or Abrikosov-Suhl) resonance in the density of states can be obtained from the Anderson Hamiltonian in the limit U CO (Lacroix 1981). For a Fermi level in the center of a valence band of width 2D and a constant density of states coupled by a constant matrix element V to an Nf-fold degenerate localized level at energy Sf below two peaks can arise in the density of states of the coupled system at low temperatures if is not too small. As... 

Anderson S studied a problem of this kind in 1958. (He had in mind disordered spin systems, but the mathematics is equivalent.) The Anderson Hamiltonian is of the general form given by Eq. (46), where Sj is uniformly distributed between jW and the off-diagonal terms are eonfined to nearest-neighbor interactions of constant magnitude J. In the limits W/J = 0 and oo it is clear that the eigenstates are extended and localized, respectively. The question is what happens at intermediate values. Anderson confined his attention to the center of... 

In the case of delocalized basis states tpa(r), the main matrix elements are those with 0 = 7 and f3 = 6, because the wave functions of two different states with the same spin are orthogonal in real space and their contribution is small. It is also true for the systems with localized wave functions tpa(r), when the overlap between two different states is weak. In these cases it is enough to replace the interacting part by the Anderson-Hubbard Hamiltonian, describing only density-density interaction... 

Fig. 21. Real part of the conductivity of YbFe4St>i2- The symbols on die left axis represent dc values at different temperatures. Below T (fv 50 K), a narrow peak at zero frequency and a gap-like feature at 18 meV gradually develop. Inset Renormalized band structure calculated from die Anderson lattice Hamiltonian. % and f denote bands of free carriers and localized electrons, respectively. At low temperatures a direct gap A opens. The Fermi level, Ep is near die top of die lower band,, resulting in hole-like character and enhanced effective mass of die quasiparticles (Dordevic et al., 2001).

In the special case where the site energies are random fluctuations, this is the Anderson model [20,21]. It is well known that Anderson used this model to prove that randomness makes a crystal become an insulating material. Anderson localization is subtly related to subdiffusion, and consequently this important phenomenon can be interpreted as a form of anomalous diffusion, in conflict with the Markov master equation that is frequently adopted as the generator of ordinary diffusion. It is therefore surprising that this is essentially the same Hamiltonian as that adopted by Zwanzig for his celebrated derivation of the van Hove and, hence, of the Pauli master equation. 

Anderson s simple model to describe the electrons in a random potential shows that localization is a typical phenomenon whose nature can be understood only taking into account the degree of randomness of the system. Using a tight-binding Hamiltonian with constant hopping matrix elements V between adjacent sites and orbital energies uniformly distributed between — W/2 and W/2, Anderson studied the modifications of the electronic diffusion in the random crystal in terms of the stability of localized states with respect to the ratio W/V. 

In one dimension, the extended nature of states is broken by the presence of any disorder irrespective of its magnitude and therefore, the states get localized. But in two and three dimensions a critical degree of disorder is needed in order to induce localization in the entire band. Anderson used a simple tight-binding hamiltonian, relevant for a periodic lattice to examine this problem. The hamiltonian has the form... 

They are equivalent to the Wannier functions used by Anderson for periodic lattices (see below) [54], Under these conditions, OMO s derived by Eqs. 42 and 43 are not strictly equivalent, simply due to the fact that the one-electron Hamiltonian of A-B is not the sum of the local Hamiltonians for A and B, considered separately. However, both types of OMO s show the same defect of locaUzation. hi addition, from a practical point of view, the OMO approach leads to much simpler calculations, as shown by Anderson [54], whereas the NMO approach is closer to the real mechanism involved in the nature of interaction and will favour the use of more reahstic molecular integrals. From now and for clarity, magnetic orbitals will be written without the prime (0 notation. 

We shall examine first the localization of an electron in a one-dimensional Anderson model. The basic set of states 11 > would then be localized around each site 1 of the periodic one-dimensional lattice. The Hamiltonian is... 

Since the pioneering work of Anderson and Mott it is known that the wave functions of disordered systems are localized. The degree of this so-called Anderson localization increases with increasing disorder. Anderson s original argument was based on a simple model Hamiltonian. In a number of cases different authors > carried out numerical studies on disordered systems still using different simplified model Hamiltonians. 

If one has a disordered polymer, one expects that some or all of the states are localized, depending on the degree of disorder. This Anderson localization, which was shown with the help of a tight-binding Hamiltonian, also takes place at the Hartree-Fock level (see Section 4.2 and the work of Day et starting at the band edges and... 

Following Anderson (1963), the model Hamiltonian of Eq. (17.12) can be deduced easily by using second quantization. Let us consider a simple model for a system of N electrons described within a basis set of N orthonormal spatial orbitals. Each electron is assumed to be localized on one orbital (site). The many-body Hamiltonian of this model is ... 

Sheng and Cooper (1995) took an alternative approach to the heavy fermion materials. The model which they use is based on the Anderson lattice, however, the on-site couplings play a crucial role. The most novel aspect of their model is the detailed recognition of how the non-spherical crystalline environment causes hybridization between the on site f electrons and ligand electrons centered off site. Their approach to the model is to first diagonalize the local parts of the Hamiltonian. For Ce based heavy fermion systems this is done in the space of local two particle states, supplemented with the trivial vacuum state 10). The periodic nature of the lattice is then re-introduced by treating the localized two electron states as forming composite particles. 

In the calculation of electronic structures, the presence of correlations thus always represents a difficulty. Perturbation expansions can account for the two extreme cases the delocalized limit in which the effective repulsion U is low compared to the band width, and the quasi-atomic limit where the electron delocalization modifies only slightly the correlated ground state (Anderson, 1959). Some variational techniques (Hubbard, 1964 Gutzwiller, 1965) allow a treatment of systems with U of the order of jS, but they are difficult to use. New methods have recently been developed for adding a part of the Hubbard Hamiltonian to the LDA (local density approximation) ground state (Czyzyk and Sawatzky, 1994). 

The interactions between a localized vibrational state with the host elastic continuum can be described by the perturbation theory of Fano or in an equivalent way, with the second quantization field theoretical method of Anderson. The Anderson-Fano Hamiltonian appropriate for xenon hydrate is ... 

It is obviously convenient to separate these two kinds of disorder. The more important type is quantitative disorder and the study of quantitatively disordered (i.e., alloy) Hamiltonians has been widely pursued. At the simplest level, the question at issue is How are we to understand the density of states and related properties outside the regions in which simple perturbation theories work This is the subject of the next section. However, a great deal of recent work has focused on a much more subtle question What can we say about the extended or localized nature of wave functions for disordered Hamiltonians This is the Anderson problem. ... 

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