Engineering flat electronic bands in quasiperiodic and fractal loop geometries

•Exact construction of eigenstates with flat and dispersive bands is reported.•Competition between translational order and growth of aperiodicity is discussed.•The effect of magnetic field on the location of flat band states is shown.•Flux tunable re-entrant behavior of the effective mass of electron is studied.

Exact construction of one electron eigenstates with flat, non-dispersive bands, and localized over clusters of various sizes is reported for a class of quasi-one-dimensional looped networks. Quasiperiodic Fibonacci and Berker fractal geometries are embedded in the arms of the loop threaded by a uniform magnetic flux. We work out an analytical scheme to unravel the localized single particle states pinned at various atomic sites or over clusters of them. The magnetic field is varied to control, in a subtle way, the extent of localization and the location of the flat band states in energy space. In addition to this we show that an appropriate tuning of the field can lead to a re-entrant behavior of the effective mass of the electron in a band, with a periodic flip in its sign.