Hyperbolic Topological Band Insulators

TL;DR

This paper introduces hyperbolic topological insulator models, revealing their non-trivial topology through invariants and demonstrating robust edge states, expanding the understanding of topological phases in hyperbolic lattices.

Contribution

It develops the first elementary models of hyperbolic topological insulators, specifically hyperbolic Haldane and Kane-Mele models, and explores their topological properties.

Findings

Non-trivial topological invariants in hyperbolic models

Evidence of bulk-boundary correspondence

Robust edge states against disorder

Abstract

Recently, hyperbolic lattices that tile the negatively curved hyperbolic plane emerged as a new paradigm of synthetic matter, and their energy levels were characterized by a band structure in a four- (or higher-)dimensional momentum space. To explore the uncharted topological aspects arising in hyperbolic band theory, we here introduce elementary models of hyperbolic topological band insulators: the hyperbolic Haldane model and the hyperbolic Kane-Mele model; both obtained by replacing the hexagonal cells of their Euclidean counterparts by octagons. Their non-trivial topology is revealed by computing topological invariants in both position and momentum space. The bulk-boundary correspondence is evidenced by comparing bulk and boundary density of states, by modelling propagation of edge excitations, and by their robustness against disorder.