Characteristics of static disorder can emerge from electron-phonon interactions

Electronic systems with large amounts of static disorder can exhibit distinct properties, including localisation of electronic states and sub-gap band tails in the density of states and electronic absorption. 

Eric Heller and collaborators have recently published a nice series of papers that show how these properties can also appear, at least on sufficiently long time scales, in the absence of disorder, due to the electron-phonon interaction. On a technical level, a coherent state representation for phonons is used. This provides a natural way of taking a classical limit, similar to what is done in quantum optics for photons. Details are set out in the following paper 

Coherent charge carrier dynamics in the presence of thermal lattice vibrations, Donghwan Kim, Alhun Aydin, Alvar Daza, Kobra N. Avanaki, Joonas Keski-Rahkonen, and Eric J. Heller

This work brought back memories from long ago when I was a postdoc with John Wilkins. I was puzzled by several related things about quasi-one-dimensional electronic systems, such as polyacetylene, that underwent a Peierls instability. First, the zero-point motion of the lattice was comparable to lattice dimerisation that produced an energy gap at the Fermi energy. Second, even in clean systems, there was a large subgap optical absorption. Third, there was no sign of the square-root singularity expected in the density of states, predicted by theories which treated the lattice classically, i.e., calculated electronic properties in the Born-Oppenheimer approximation.

I found that on the energy scales relevant to the sub-gap absorption, the phonons could be treated like static disorder and make use of known exact results for one-dimensional Dirac equations with random disorder. This explained the puzzles.

Effect of Lattice Zero-Point Motion on Electronic Properties of the Peierls-Fröhlich State

The disorder model can also be motivated by considering the Feynman diagrams for the electronic Green's function perturbation expansion in powers of the electron-phonon interaction. In the limit that the phonon frequency is small, all the diagrams become like those for a disordered system, where the strength of the static disorder is given by 

I then teamed up with another postdoc, Kihong Kim, who calculated the optical conductivity for this disorder model.

Universal subgap optical conductivity in quasi-one-dimensional Peierls systems

Two things were surprising about our results. First, the theory agreed well with experimental results for a range of materials, including the temperature dependence. Second,  the frequency dependence had a universal form. Wilkins was clever and persistent at extracting such forms, probably from his experience working on the Kondo problem.