Haldane model on the Sierpi\'nski gasket

TL;DR

This paper explores the topological phases of the Haldane model on a fractal Sierpiński gasket, revealing complex phase diagrams and topological states influenced by fractal geometry, with real-space invariants and robustness to disorder.

Contribution

It introduces a real-space Chern number approach for topological characterization on fractals and demonstrates the impact of fractality on the phase diagram of the Haldane model.

Findings

Multiple fractal gaps and a flat band appear due to fractal geometry.

Topological states are robust against disorder.

Complex phase diagrams emerge for a single Sierpiński gasket, differing from previous doubled systems.

Abstract

We investigate the topological phases of the Haldane model on the Sierpi\'nski gasket. As a consequence of the fractal geometry, multiple fractal gaps arise. Additionally, a flat band appears, and due to a complex next-nearest neighbour hopping, this band splits and multiple topological flux-induced gaps emerge. Owing to the fractal nature of the model, conventional momentum-space topological invariants cannot be used. Therefore, we characterise the system's topology in terms of a real-space Chern number. In addition, we verify the robustness of the topological states to disorder. Finally, we present phase diagrams for both a fractal gap and a flux-induced gap. Previous work on a similar system claims that fractality "squeezes" the well-known Haldane phase diagram. However, this result arises because a doubled system was considered with two Sierpi\'nski gaskets glued together. We consider only a single copy of the Sierpi\'nski gasket, keeping global self-similarity. In contrast with these previous results, we find intricate and complex patterns in the phase diagram of this single fractal. Our work shows that the fractality of the model greatly influences the phase space of these structures, and can drive topological phases in the multitude of fractal and flux-induced gaps, providing a richer platform than a conventional integer dimensional geometry.