Disorder perturbed Flat Bands II: a search for criticality

We seek the possibility of a disorder driven transition in a tight-binding lattice with a flat band using complexity parameter approach. Our results indicate the existence of a localized to extended states transition with increasing disorder, insensitive to disorder strength, in weak disorder limit; the spectral statistics at the critical point corresponds to a critical Brownian ensemble, a non-equilibrium universality class of random matrix ensembles, intermediate to Poisson and Gaussian orthogonal ensemble. With increasing disorder, the statistics again approaches Poisson limit indicating a localization -> extended -> localization transition of the wave-dynamics. Our analysis also reveals a hidden connection of weakly disordered flat bands to a wide-range of other complex systems including standard Anderson Hamiltonian.