Non-Commutative Chern Numbers for Generic Aperiodic Discrete Systems

TL;DR

This paper extends the concept of Chern numbers to aperiodic systems, providing a theoretical framework and numerical evidence for quantized topological invariants in amorphous and quasi-crystalline materials.

Contribution

It introduces a unifying operator theoretic approach to define non-commutative Chern numbers for aperiodic lattices, ensuring their quantization and stability.

Findings

Numerical evidence of quantized Hall conductance in amorphous lattices.

Extension of Chern number formulas to non-canonical lattice Hamiltonians.

Applicability to a wide range of aperiodic and synthetic systems.

Abstract

The search for strong topological phases in generic aperiodic materials and meta-materials is now vigorously pursued by the condensed matter physics community. In this work, we first introduce the concept of patterned resonators as a unifying theoretical framework for topological electronic, photonic, phononic etc. (aperiodic) systems. We then discuss, in physical terms, the philosophy behind an operator theoretic analysis used to systematize such systems. A model calculation of the Hall conductance of a 2-dimensional amorphous lattice is given, where we present numerical evidence of its quantization in the mobility gap regime. Motivated by such facts, we then present the main result of our work, which is the extension of the Chern number formulas to Hamiltonians associated to lattices without a canonical labeling of the sites, together with index theorems that assure the quantization and stability of these Chern numbers in the mobility gap regime. Our results cover a broad range of applications, in particular, those involving quasi-crystalline, amorphous as well as synthetic (i.e. algorithmically generated) lattices.