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In the context of Anderson Localization, I heard statements such as the following: "Due to disorder, there is a broadening of the bands. Although spectral gaps between continuous bands may shrink or even vanish, one still has mobility gap between the continuous spectra."

As far as I understood, a mobility gap refers to an interval in the spectrum of a Hamiltonian, which contains only pure point spectrum and for which the corresponding eigenfunctions have a certain localization/decay behavior.

Question: Is there a strict definition of the term mobility gap and how does it connect to transport properties (in particular, conductivity)?

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Usually metals and insulators are distinguished by seeing whether the Fermi energy cuts through the spectrum (i. e. the material is a conductor aka a “metal”) or lies in a spectral gap (i. e. the material is an insulator or a semiconductor). In the metallic case, you have non-zero conductivity, in the insulating case, you have no conductivity.

Anderson localization explains why in disordered media the system may be insulating (zero conductivity) even though the Fermi energy cuts through the spectrum: here, the absence of conductivity is due to the nature of the spectrum, which is dense, pure point spectrum. That means to each energy $E_{\mathrm{F}}$ inside the region of spectral localization you can find an eigenfunction with an eigenvalue $E$ that is arbitrarily close to $E_{\mathrm{F}}$. Because eigenstates are localized (they are like bound states in the hydrogen atom), they carry no current. In contrast, states in the continuous spectrum are delocalized (the analog of ionized states in the hydrogen atom) and are therefore able to carry current.

While in such systems there is no spectral gap, the two conducting regions are separated by a mobility gap, because two conducting spectral regions are separated in energy from one another by a region consisting only of bound states.

Anderson localization is a wave phenomenon that is due to destructive interference: in a purely periodic system, you can have constructive and destructive interference by the regular structure. In a disordered system, it is much harder to have constructive interference, though.

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