Fractality-induced Topology

TL;DR

This paper introduces a theoretical framework using isospectral reduction to reveal topological phases in fractal geometries, highlighting the potential for naturally occurring topological states in fractal materials.

Contribution

The work demonstrates that fractals can host topological phases without traditional mechanisms, providing a new approach to analyze complex self-similar structures.

Findings

Fractals can support topologically protected boundary and corner states.

Isospectral reduction effectively simplifies fractal structures to reveal topological features.

Topological phases may naturally occur in fractal-structured materials.

Abstract

Fractal geometries, characterized by self-similar patterns and non-integer dimensions, provide an intriguing platform for exploring topological phases of matter. In this work, we introduce a theoretical framework that leverages isospectral reduction to effectively simplify complex fractal structures, revealing the presence of topologically protected boundary and corner states. Our approach demonstrates that fractals can support topological phases, even in the absence of traditional driving mechanisms such as magnetic fields or spin-orbit coupling. The isospectral reduction not only elucidates the underlying topological features but also makes this framework broadly applicable to a variety of fractal systems. Furthermore, our findings suggest that these topological phases may naturally occur in materials with fractal structures found in nature. This work opens new avenues for designing fractal-based topological materials, advancing both theoretical understanding and experimental exploration of topology in complex, self-similar geometries.