Instanton calculus of Lifshitz tails

Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Some degree of quenched disorder is present in nearly all solids, and can have a marked impact on their macroscopic properties. A manifestation of this effect is the Lifshitz tail of localized states that then gets attached to the energy spectrum, resulting in the nonzero density of states in the band gap. We present here a systematic approach for deriving the asymptotic behavior of the density of states and of the typical shape of the disorder potentials in the Lifshitz tail. The analysis is carried out first for the well-controlled case of noninteracting particles moving in a Gaussian random potential and then for a broad class of disordered scale-invariant models—pertinent to a variety of systems ranging from semiconductors to semimetals to quantum critical systems. For relevant Gaussian disorder, we obtain the general expression for the density of states deep in the tail, with the rate of exponential suppression governed by the dynamical exponent and spatial dimensions. For marginally relevant disorder, however, we would expect a power-law scaling. We discuss the implications of these results for understanding conduction in disordered materials.