  We unravel a fundamental connection between supersymmetry and a wide class of
two dimensional second-order topological insulators (SOTI). This particular
supersymmetry is induced by applying a half-integer Aharonov-Bohm flux
$f=\Phi/\Phi_0=1/2$ through a hole in the system. Here, three symmetries are
essential to establish this fundamental link: chiral symmetry, inversion
symmetry, and mirror symmetry. At such a flux of half-integer value the mirror
symmetry anticommutes with the inversion symmetry leading to a nontrivial
$n=1$-SUSY representation for the absolute value of the Hamiltonian in each
chiral sector, separately. This implies that a unique zero-energy state and an
exact twofold degeneracy of all eigenstates with non-zero energy is found even
at finite system size. For arbitrary smooth surfaces the link between 2D-SOTI
and SUSY can be described within a universal low-energy theory in terms of an
effective surface Hamiltonian which encompasses the whole class of
supersymmetric periodic Witten models. Applying this general link to the
prototypical example of a Bernevig-Hughes-Zhang(BHZ)-model with an in-plane
Zeeman field, we analyze the entire phase diagram and identify a gapless Weyl
phase separating the topological from the non-topological gapped phase.
Surprisingly, we find that topological states localized at the outer surface
remain in the Weyl phase, whereas topological hole states move to the outer
surface and change their spatial symmetry upon approaching the Weyl phase.
Therefore, the topological hole states can be tuned in a versatile manner
opening up a route towards magnetic-field-induced topological engineering in
multi-hole systems. Finally, we demonstrate the stability of localized states
against deviation from half-integer flux, flux penetration into the sample,
surface distortions, and random impurities for impurity strengths up to the
order of the surface gap.
