Lifshitz tails and long-time decay in random systems with arbitrary disorder |
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Authors: | J. M. Luck Th. M. Nieuwenhuizen |
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Affiliation: | (1) Service de Physique Théorique, CEN Saclay, 91191 Gif-sur-Yvette, France;(2) Institut für Theoretische Physik, RWTH, 5100 Aachen, Federal Republic of Germany |
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Abstract: | In random systems, the density of states of various linear problems, such as phonons, tight-binding electrons, or diffusion in a medium with traps, exhibits an exponentially small Liftshitz tail at band edges. When the distribution of the appropriate random variables (atomic masses, site energies, trap depths) has a delta function at its lower (upper) bound, the Lifshitz singularities are pure exponentials. We study in a quantitative way how these singularities are affected by a universal logarithmic correction for continuous distributions starting with a power law. We derive an asymptotic expansion of the Lifshitz tail to all orders in this logarithmic variable. For distributions starting with an essential singularity, the exponent of the Lifshitz singularity itself is modified. These results are obtained in the example of harmonic chains with random masses. It is argued that analogous results hoid in higher dimensions. Their implications for other models, such as the long-time decay in trapping problems, are also discussed. |
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Keywords: | Random harmonic chains Lifshitz singularities trapping problems density of states |
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