A Universal Chern Model on Arbitrary Triangulations

Abstract

Given a triangulation of a closed orientable surface, we consider the lattice with sites at the vertices, edges and facets of the triangulation. Borrowing from mathematics literature, we introduce on this lattice a pair of tight-binding Hamiltonians derived from the boundary and Poincaré duality maps of finite simplicial manifolds. These Hamiltonians have been proved to have 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.

Figures (7)

• Figure 1: Left: Stanford Bunny, a triangulated mesh containing 69,451 faces generated by a 3D scan of a ceramic figurine Bunny. Right: A simplified version containing 1549 vertices, 4641 edges and 3094 faces, to be used in our computer simulations.
• Figure 2: A region of a triangulation and its associated data, consisting of labels and orientations of the simplices. The marked vertices and the seen orientations order the vertices of the faces as $|\tau_1\rangle = |\textcolor{red}{\pi_1}\pi_2\pi_3\rangle$ and $|\tau_2\rangle = |\textcolor{red}{\pi_4}\pi_3 \pi_2 \rangle$. The triangulation is populated with single-mode resonators (the disks) in a one-to-one fashion with its simplices.
• Figure 3: Resonator couplings needed to implement the operator $B+B^\dagger$ (continuous thick lines), and operator $S$ (oriented dashed lines), for the data from Fig. \ref{['Fig:TM']}. The color coding and other conventions are explained in Fig. \ref{['Fig:CR']}.
• Figure 4: Implementation of the hopping terms of a tight-binding Hamiltonian using bridged acoustic cavities (C) with fundamental frequency $\omega_0$. Each panel shows the physical connections (top), together with their symbols (middle) and the hopping terms (bottom) they implement. Panel (c) reduces a purely imaginary hopping term to the connections from panels (a) and (b), at the expense of introducing copies of the resonators (see main text). The value of $w = \tfrac{1}{2}(\omega_S^2 - \omega_A^2)$ can be adjusted by varying the cross section of the bridges. The coloring of the cavities is a rough representation of the air pressure field.
• Figure 5: The bulk spectra of the indicated operators when deployed on surfaces of different genus $g$. The number of zero modes of $B+B^\dagger$ confirms the relation \ref{['Eq:ZeroModes']}. The features on these surfaces are placeholders for the resonators.