Long range disorder and Anderson transition in systems with chiral symmetry

We study the spectral properties of a chiral random banded matrix (chRBM) with elements decaying as a power-law Hij ∼ vertical bar i-j vertical bar -α. This model is equivalent to a chiral 1D Anderson Hamiltonian with long range power-law hopping. In the weak disorder limit we obtain explicit nonperturbative analytical results for the density of states (DoS) and the two-level correlation function (TLCF) by mapping the chRBM onto a nonlinear σ model. We also put forward, by exploiting the relation between the chRBM at α=1 and a generalized chiral random matrix model, an exact expression for the above correlation functions. We give compelling analytical and numerical evidence that for this value the chRBM reproduces all the features of an Anderson transition. Finally we discuss possible applications of our results to quantum chromodynamics (QCD)