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

TL;DR

This paper reveals a fundamental link between supersymmetry and second-order topological insulators in 2D, showing how flux, symmetries, and surface states interplay to produce novel topological phases and states.

Contribution

It establishes a universal connection between supersymmetry and 2D second-order topological insulators using symmetry analysis and effective surface Hamiltonians.

Findings

Supersymmetry arises at half-integer flux in 2D topological insulators.

Topological states persist in the Weyl phase, enabling topological engineering.

Localized states are stable against various perturbations.

Abstract

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.