We study the band-centre anomaly in the one-dimensional Anderson model with weak correlated disorder. Our analysis is based on the Hamiltonian map approach; the correspondence between the discrete model and its continuous counterpart is discussed in
detail. We obtain analytical expressions of the localisation length and of the invariant measure of the phase variable, valid for energies in a neighbourhood of the band centre. By applying these general results to specific forms of correlated disorder, we show how correlations can enhance or suppress the anomaly at the band centre.
We consider heat transport in one-dimensional harmonic chains with isotopic disorder, focussing our attention mainly on how disorder correlations affect heat conduction. Our approach reveals that long-range correlations can change the number of low-f
requency extended states. As a result, with a proper choice of correlations one can control how the conductivity $kappa$ scales with the chain length $N$. We present a detailed analysis of the role of specific long-range correlations for which a size-independent conductivity is exactly recovered in the case of fixed boundary conditions. As for free boundary conditions, we show that disorder correlations can lead to a conductivity scaling as $kappa sim N^{varepsilon}$, with the scaling exponent $varepsilon$ being arbitrarily small (although not strictly zero), so that normal conduction is almost recovered even in this case.
