Kane-Mele with a twist: Quasicrystalline higher-order topological insulators with fractional mass kinks

We establish an analytic low-energy theory describing higher-order topological insulator (HOTI) phases in quasicrystalline systems. We apply this to a model consisting of two stacked Haldane models with oppositely propagating edge modes, analogous to the Kane-Mele model, and with a $30^\circ$ twist. We show that the resulting localized modes at corners, characteristic of a HOTI, are not associated with conventional mass inversions but are instead associated with what we dub "fractional mass kinks". By generalizing the low-energy theory, we establish a classification for arbitrary $n$-fold rotational symmetries. We also derive a relationship between corner modes in a bilayer and disclination modes in a single layer. By using numerics to go beyond the weak-coupling limit, we show that a hierarchy of additional gaps occurs due to the quasiperiodicity, which also harbor corner-localized modes.

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