Chiral-symmetry breaking and degeneracy in Koch fractal geometries

Abstract

Fractal geometries provide a distinctive platform for controlling quantum interference, topology, and localization. We investigate the Su--Schrieffer--Heeger (SSH) model constructed on the Koch curve, where a triangular geometry enables additional same-sublattice hoppings that break chiral symmetry and modify the conventional topological behavior captured by a real-space local marker. Destructive interference within the triangular units produces a highly degenerate flat level at $E=-t$. Although this level originates from the local geometry, its degeneracy and spectral weight are controlled by the self-similar connectivity of the Koch curve. The hierarchy reduces the number of independent flat states and reorganizes the spectrum into an intricate distribution of energy levels. Our results reveal complementary roles for local triangular interference and global fractal connectivity in shaping the electronic properties of geometrically modified SSH chains.