The one-dimensional Anderson model: scaling and resonances revisited

Abstract

Using the finite-size scaling theory of Fisher and Barber (1972), which is shown to be equivalent to the scaling theory of Abrahams and co-workers (1979), the author investigates the crossover from the linear increase of the residual quantum resistance to the exponential one, as a function of the size of the system. Starting from an explicit tight-binding Hamiltonian with diagonal disorder and using Landauer's formula, he relates the resistance to the trace of a random unimodular matrix product whose eigenvalues are extensively studied. Special care is devoted to length scales smaller than the localisation length, where a classical ohmic behaviour is usually assumed, characterised by a resistivity proportional to the inverse localisation length. He shows numerically that this scaling behaviour is masked by giant fluctuations. He explains, in a simple and pedagogical case, how the actual behaviour of the resistance differs from the scaling prediction by resonance effects. Nevertheless, by summing the fluctuating eigenvalues which are studied in this work for increasing values of the length of the disordered specimen, he gives a more convincing test of the validity of the finite-size concept in one-dimensional Anderson models.