The Anderson Problem

For a disordered Hamiltonian it is natural to ask what the nature of the wave functions is. In particular, do they extend throughout the solid or are they localized It is important to note that this is a meaningful question even in the independent-electron approximation and indeed all of this section is in that spirit. What do we mean by localization Thouless has listed six definitions which have been used, with the statement that they are probably equivalent. Three of these are given below. 

The wave functions at the Fermi level are localized if the dc conductivity vanishes (for a static lattice) and the ac conductivity (t(co) is of order (o. This is the most important practical consequence of localization (Section 7.6). 

A change of boundary conditions shifts levels corresponding to localized states by an amount of order exp( — rather than This is a very useful condition in numerical studies. In practice, the boundary conditions are changed from periodic to antiperiodic. 

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 

Recent numerical work by Kirkpatrick and Eggarter has shown that in lattices with two interpenetrating sublattices (square, simple cubic, bcc, etc.) localized states can occur at the center of the band while extended states still survive elsewhere, for random binary alloy Hamiltonians. This is in contradiction to some of the conjectured descriptions of localization that have been given, and is indicative of the rather confused state in which this field remains. 

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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. ... 

The study of the nature of states in band tails (the Anderson problem, Section 7.4) has not yet progressed to the point where detailed quantitative comparison with experiment are meaningful. So far the emphasis has been on the extraction of qualitative and semiquantitative consequences of the localization of such states. Mott concluded that, if conduction was due to phonon-assisted hopping among such localized states, it should vary with temperature... 

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