  Recently, log-periodic quantum oscillations have been detected in topological
materials zirconium pentatelluride (ZrTe5) and hafnium pentatelluride (HfTe5),
displaying intriguing discrete scale invariance (DSI) characteristic. In
condensed materials, the DSI is considered to be related to the quasi-bound
states formed by massless Dirac fermions with strong Coulomb attraction,
offering a feasible platform to study the long-pursued atomic-collapse
phenomenon. Here, we demonstrate that a variety of atomic vacancies in the
topological material HfTe5 can host the geometric quasi-bound states with DSI
feature, resembling the artificial supercritical atom collapse. The density of
states of these quasi-bound states are enhanced and the quasi-bound states are
spatially distributed in the "orbitals" surrounding the vacancy sites, which
are detected and visualized by low-temperature scanning tunneling
microscope/spectroscopy (STM/S). By applying the perpendicular magnetic fields,
the quasi-bound states at lower energies become wider and eventually invisible,
meanwhile the energies of quasi-bound states move gradually towards the Fermi
energy (EF). These features are consistent with the theoretical prediction of a
magnetic-field-induced transition from supercritical to subcritical states. The
direct observation of geometric quasi-bound states sheds light on the deep
understanding of the DSI in quantum materials.
