On Mott’s formula for the ac-conductivity in the Anderson model

Abstract

We study the ac-conductivity in linear response theory in the general framework of ergodic magnetic Schrödinger operators. For the Anderson model, if the Fermi energy lies in the localization regime, we prove that the acconductivity is bounded from above by Cν 2 (log 1 ν ) d+2 at small frequencies ν. This is to be compared to Mott's formula, which predicts the leading term to be Cν 2 (log 1 ν ) d+1 . IntroductionThe occurrence of localized electronic states in disordered systems was first noted by Anderson in 1958 [An], who argued that for a simple Schrödinger operator in a disordered medium,"at sufficiently low densities transport does not take place; the exact wave functions are localized in a small region of space." This phenomenon was then studied by Mott, who wrote in 1968 [Mo1]: "The idea that one can have a continuous range of energy values, in which all the wave functions are localized, is surprising and does not seem to have gained universal acceptance." This led Mott to examine Anderson's result in terms of the Kubo-Greenwood formula for σ EF (ν), the electrical alternating current (ac) conductivity at Fermi energy E F and zero temperature, with ν being the frequency. Mott used its value at ν = 0 to reformulate localization: If a range of values of the Fermi energy E F exists in which σ EF (0) = 0, the states with these energies are said to be localized; if σ EF (0) = 0, the states are nonlocalized.Mott then argued that the direct current (dc) conductivity σ EF (0) indeed vanishes in the localized regime. In the context of Anderson's model, he studied the behavior of Re σ EF (ν) as ν → 0 at Fermi energies E F in the localization region (note Im σ EF (0) = 0). The result was the well-known Mott's formula for the ac-conductivity at zero temperature [Mo1], [Mo2], which we state as in

Smoothness of Correlations in the Anderson Model at Strong Disorder

“…, and a recent result by Klein, Lenoble, and Müller [13] proved an upper bound on the AC conductivity in the spirit of Mott's formula. This has the form…”

“…for a well-defined conductivity measure Σ E (see [13]). If the current-current correlation function has a density m(…”

“…In order to prove analyticity of the correlation functions G A (z), as expressed in (14), we first need to analyze the Cauchy-type integrals J n (h; z) defined in (13). We begin with estimates on the simplest form of these integrals for which n and z depend on one variable only.…”

Microscopic foundations of Ohm and Joule's laws – The relevance of thermodynamics

“…In 2007, A. Klein, O. Lenoble and P. Müller introduced [20] for the first time the concept of a "conductivity measure" for a system of non-interacting fermions subjected to a random potential. More precisely, the authors considered the Anderson tight-binding model in presence of a time-dependent spatially homogeneous electric field that is adiabatically switched on.…”

“…The fermionic nature of charge carriers -electrons or holes in crystals -as well as thermodynamics of such systems were implemented by choosing the Fermi-Dirac distribution as the initial density matrix of particles. In [20] only systems at zero temperature with Fermi energy lying in the localization regime are considered, but it is shown in [21] that a conductivity measure can also be defined without the localization assumption and at any positive temperature. Their study can thus be seen as a mathematical derivation of Ohm's law for space-homogeneous electric fields having a specific time behavior.…”