Quasicrystalline Analogue of the Haldane Model

TL;DR

This paper introduces a topological quasicrystalline model analogous to the Haldane model, demonstrating topological phases, Dirac cones, and potential for strongly-correlated physics in a quasicrystalline setting suitable for cold-atom experiments.

Contribution

It develops a quasicrystalline topological model with complex couplings, analyzes its bandstructure, and maps its phase diagram, extending topological concepts to quasiperiodic systems.

Findings

Presence of symmetry-protected Dirac cones gapped by TRS-breaking terms

Identification of a topological phase with Chern number 1

Discovery of narrow Chern bands with potential for correlated physics

Abstract

We present a model for a topological quasicrystalline system which is suitable for realisation in cold-atom experiments. We define the model in terms of complex momentum-space couplings which break time-reversal symmetry (TRS), and detail how it may be experimentally realised using two-photon Raman couplings. In the weak-potential limit, we study the model analytically by calculating the bandstructure over a `quasi-Brillouin zone' (QBZ). We find symmetry-protected Dirac cones, which are gapped by a TRS-breaking term, resulting in a Chern number $\mathcal{C}=1$. This provides a direct analogy to the Haldane model, but now in a quasicrystalline setting. We also infer the number of states below the topological gap from the QBZ area. We verify our analysis with numerical calculations of periodic approximants to our system, constructing a phase diagram in parameter space which shows a topological region extending beyond the weak-potential regime. We also find examples of narrow Chern bands with the potential for hosting strongly-correlated physics. Our work raises questions about the nature of localisation and strongly-correlated states in Chern bands in quasiperiodic systems.