Localization of Normal Modes and Energy Transport in the Disordered Harmonic Chain

Abstract

The feature of normal modes in one-dimensional isotopically disordered harmonic chain is investigated. It is proved for any frequency that the simultaneous difference equations for the displacement u_n of the n-th atom, (-∞ ≪ n ≪ ∞), has a solution in which u_n grows exponentially with n with probability 1. In the low frequency limit the rate of exponential growth γ is explicitly calculated. A necessary and sufficient condition is obtained for there to exist a localized solution such that lim _{n → ±∞}u_n = 0. It is proved that any infinite disordered chain, except those with measure 0, can be made to have an exponentially localized solution for any ω by modifying the mass of one atom m₀ to a suitable real value as a function of ω. From this it does not logically follow that almost all normal modes of any given large but funite sample chain are localized in such a way as occurred in the above modified infinite chain. However, the theoretical estimate of the nature of normal modes and energy transport based on the above mentioned value of r, agrees well with the result of computer experiments. As a by-product of our investigation, the exact expression of the thermal conductivity due to the Kubo formula for the isotopically disordered harmonic chain is obtained in a closed form.