Scaling theory of localization

1-Anderson localization:

Before 1958, there was an old view about the behavior of an electronic system when we add a random potential. According to this view, the only effect of randomness on an electronic system is that the Bloch waves lose phase coherence on the length scale of the mean free path l, but in general the wave function stays extended throughout the sample. In 1958 Anderson proposed that the electronic systems maybe altered significantly if the disorder level is high. That means if we add a strong random potential the wave function may become localized and so the wave function decays exponentially around the random center in space,i.e., $\mid \psi ({\overrightarrow {r}})\mid \sim exp(\mid {\overrightarrow {r}}-{\overrightarrow {r}}_{0}\mid /\xi )$

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \xi} is the localization length. In the strong disorder limit, and in the zero-order description we can imagine most of the eigenstates are localized around the impurity sites and then try to sum up all wave functions linearly as a perturbation to the system. We could see the admixture between different orbitals will not produce an extended state because the orbitals are nearby have large different energy and the orbitals that have the same energy are far apart in space and in both cases next order perturbation has small value and does not affect the localized states significantly.

In one dimension, all states are localized and it is not important how weak the disorder is. In two dimensions, they have the same behavior but in three dimensions it depends on the disorder level. That means we have a transition between metal to insulator if we increase the disorder level, but we have still extended states in the weak disorder. The dividing line between localized and extended states is called the mobility edge that is a function of energy $\varepsilon$ and disorder strength $w$.

Here we introduce the localization lenght as \xi^{-1}\equiv\underset{m\rightarrow\infty}{lim}\frac{-1}{2m}Ln\overline{\mid<0\mid G\mid m>\mid^{2}}

Where the bar means the average over the random potentials on each site, and G is the Green's function for system. Why we introduce the lcoalization length in this form? Assume the disorder site has energy $\varepsilon$ and the wave function falls off exponentially as $exp({\frac {-m}{\xi }}$), where $m$ is the distance from the peak. So we can calculate Green's function as below:

$<m\mid G(\varepsilon -i\eta )\mid 0>=\sum _{n}{\frac {<m\mid \varepsilon _{n}><\varepsilon _{n}\mid 0>}{\varepsilon -\varepsilon _{n}-i\eta }}$

$<m\mid G(\varepsilon -i\eta )\mid 0>=V\int d\varepsilon ^{'}D(\varepsilon ^{'}){\frac {<m\mid \varepsilon _{n}><\varepsilon _{n}\mid 0>}{\varepsilon -\varepsilon _{n}-i\eta }}\thickapprox {\frac {V\xi }{N}}D(\varepsilon )i\pi <m\mid \varepsilon >$

Where N is the number of sites in th elattice. The wave function has width $\xi$ so a fraction ${\frac {\xi }{N}}$ has overlap of order unity with site 0 ,and all the rest can be neglected. So, we have

$<m\mid G(\varepsilon -i\eta )\mid 0>\thickapprox {\frac {V\xi }{N}}D(\varepsilon )i\pi e^{i\phi }e^{-{\frac {m}{\xi }}}$

Where $\phi$ describes the phase of the matrix element. As you can see by using of the definition for the localization length we can get good result for $\xi$ because we we calculate average over all random possibilities the phase$(\phi )$ goes to zero.

2- Scaling Theory of Localization:

This method is the most applicable method to describe localization that was proposed by Abrahams et.al.(1979). According to this theory, there is a general behavior for materials so that if we can measure the resistance for a sample then we can predict the resistance of a sample of any other length or that sample. That means the resistance of a sample follows this relationship: $R(l)=R({\frac {l}{l_{0}}})$ where l_{0}is the length of sample that its resistance is equal to the basic quantum of resistance $R_{H}:R_{H}\equiv {\frac {h}{e^{2}}}=25.813\Omega$.

In another notation we can say there is a function for the conductance as $g=F({\frac {L}{\xi }})$ that the function F is called scaling function. In this definition the correlation length $(\xi$) includes all microscopic length scales such as the Fermi wave number,mean free path etc, so this function is a single parameter scaling and these scales do not enter explicitly in the formula.