A Universal Chern Model on Arbitrary Triangulations

TL;DR

This paper introduces a universal topological model on arbitrary surface triangulations that produces robust Chern insulator phases, confirmed through simulations and realizable in metamaterials for practical applications.

Contribution

It develops a universal Chern model applicable to any triangulated surface, enabling topological phases and edge modes in diverse geometries and physical systems.

Findings

The model exhibits large, clean topological gaps with non-trivial Chern numbers.

Numerical simulations confirm the robustness of topological edge modes.

A metamaterial implementation reproduces the model's dynamics, demonstrating practical feasibility.

Abstract

Given a triangulation of a closed orientable surface, we place single-mode resonators or single-orbital artificial atoms at its vertices, edges and facets, and we devise near-neighbor hopping terms derived from the boundary and Poincar\'e duality maps of the simplicial complex of the triangulation. Regardless of the surface or its triangulation, these terms always lead to tight-binding Hamiltonians with large and clean topological spectral gaps, carrying non-trivial Chern numbers in the limit of infinite refinement of the triangulation. We confirm this via numerical simulations, and demonstrate how these models enable topological edge modes at the surfaces of real-world objects. Furthermore, we describe a metamaterial whose dynamics reproduces that of the proposed model, thus bringing the topological metamaterials closer to real-world applications.