Second-order topology and supersymmetry in two-dimensional topological insulators

Abstract

We unravel a fundamental connection between supersymmetry (SUSY) and a wide class of two-dimensional (2D) second-order topological insulators (SOTI). This particular supersymmetry is induced by applying a half-integer Aharonov-Bohm flux $f=\mathrm{\ensuremath{\Phi}}/{\mathrm{\ensuremath{\Phi}}}_{0}=1/2$ 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 nonzero energy is found even at a 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 nontopological 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 toward magnetic-field-induced topological engineering in multihole 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.