  The explicit construction of non-dispersive flat band modes and the
tunability of has been reported for a hierarchical 3-simplex fractal geometry.
A single band tight binding Hamiltonian defined for the deterministic
self-similar non-translationally invariant network can give rise to a countably
infinity of such self localized eigenstates for which the wave packet gets
trapped inside a characteristic cluster of atomic sites. An analytical
prescription to detect those dispersionless states has been demonstrated
elaborately. The states are localized over clusters of increasing sizes,
displaying the existence of a multitude of localization areas. The onset of
localization can, in principle, be delayed in space by an appropriate choice of
the energy of the electron. The response of the system with the modulation of
the anisotropy parameter is also studied. Supportive calculation of spectral
landscape and demonstration of band dispersion plot are presented to solidify
the analytical results. Variation of effective mass tensor cites re-entrant
behavior with respect to the modulation of off-diagonal anisotropy. The
tunability of those states leads to the controlled decay of wave function
envelope. The impact of uniform magnetic perturbation on the bound states has
also been discussed. Continuous variation of flux modulates the position of the
flat band modes. The macroscopic degeneracy associated with the modes is
retained with respect to the application of perturbation.
