Flat electronic bands in fractal-kagomé network and the effect of perturbation

We demonstrate an analytical prescription of demonstrating the flat band [FB] states in a fractal incorporated kagome type network that can give rise to a countable infinity of flat non-dispersive eigenstates with a multitude of localization area. The onset of localization can, in principle, be delayed in space by an appropriate choice of energy regime. The length scale, at which the onset of localization for each mode occurs, can be tuned at will following the formalism developed within the framework of real space renormalization group. This scheme leads to an exact determination of energy eigenvalue for which one can have dispersionless flat electronic bands. Furthermore, we have shown the effect ofuniform magnetic field for the same non-translationally invariant network model that has ultimately led to an‘apparent invisibility’ of such staggered localized states and to generate absolutely continuous sub-bands in the energy spectrum and again an interesting re-entrant behavior of those FB states.