Correlated random band matrices: Localization-delocalization transitions

We study the statistics of eigenvectors in correlated random band matrix models. These models are characterized by two parameters, the bandwidth $B\left(N\right)\mathrm{}$ of a Hermitian $N\times N$ matrix and the correlation parameter $C\left(N\right)\mathrm{}$ describing correlations of matrix elements along diagonal lines. The correlated band matrices show a much richer phenomenology than models without correlation as soon as the correlation parameter scales sufficiently fast with matrix size. In particular, for $B\left(N\right)\mathrm{}\sim \sqrt{N}$ and $C\left(N\right)\mathrm{}\sim \sqrt{N},$ the model shows a localization-delocalization transition of the quantum Hall type.