Logarithmic frequency dependence of the dynamical conductivity of one-dimensional disordered systems at low frequencies: a computer simulation for a very long chain

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Abstract
The low-frequency behaviour of the dynamical conductivity sigma ( omega ) for electrons on very long (106 sites) one-dimensional disordered chains is investigated by computer simulation. Berezinskii's result (1974) which is asymptotically exact in the limit of weak disorder and shows the behaviour sigma ( omega )/ sigma 0 to 32( omega tau )2(ln2(4 omega tau )- pi 2/4+(2C-3)ln(4 omega tau )+ ...) at low frequencies ( sigma 0, tau and C denote the Drude value of the conductivity, the relaxation time of the electrons and Euler's constant, respectively), is established for the first time by a direct numerical method. A discussion is given of a simple derivation of the coefficient 32. It is also found that this low-frequency behaviour remains almost when disorder is introduced, unchanged even when the disorder is strong.