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Exact construction of one electron eigenstates with flat, non-dispersivebands, and localized over clusters of various sizes is reported for a class ofquasi-one dimensional looped networks. Quasiperiodic Fibonacci and Berkerfractal geometries are embedded in the arms of the loop threaded by a uniformmagnetic flux. We work out an analytical scheme to unravel the localized singleparticle states pinned at various atomic sites or over clusters of them. Themagnetic field is varied to control, in a subtle way, the extent oflocalization and the location of the flat band states in energy space. Inaddition to this we show that, an appropriate tuning of the field can lead to are-entrant behavior of the effective mass of the electron in a band, with aperiodic flip in its sign.

Engineering flat electronic bands in quasiperiodic and fractal loop geometries