Fractal Topology of Majorana Bound States in Superconducting Quasicrystals

TL;DR

This paper reveals that Majorana Bound States in superconducting quasicrystals exhibit a fractal topological phase diagram, influenced by the interplay between quasicrystallinity and superconducting pairing, leading to a hierarchy of their stability.

Contribution

It introduces the concept of a fractal topological phase diagram, called Majorana's Butterfly, in superconducting quasicrystals, extending the understanding of topological phases in fractal spectra.

Findings

Identification of a fractal topological phase diagram called Majorana's Butterfly.

Demonstration that the stability of Majorana states depends on the competition between quasicrystallinity and superconductivity.

Revealing a hierarchy of Majorana stability governed by energy scale competition.

Abstract

Quasicrystalline order induces a fractal energy spectrum, yet its impact on topological protection remains an open fundamental question. Here, we demonstrate that the topological phase transitions characterised by the appearance of Majorana Bound States themselves have a fractal character. By extending this analysis to the full family of Sturmian words, we uncover Kitaev's Butterfly $-$ a spectral fractal analogous to Hofstadter's butterfly, but fundamentally distinguished by a central superconducting gap. Within this framework, we identify Majorana's Butterfly as a fractal topological phase diagram governed by the competition between quasicrystallinity and superconducting pairing. We show that this competition dictates a hierarchy of Majorana stability, where the survival of the topological phase against fractal fragmentation is determined by the relative strength of these competing energy scales.