Topological linear response of hyperbolic Chern insulators

TL;DR

This paper derives a hyperbolic analog of the TKNN formula linking electromagnetic Hall response to topological invariants, simplifying the calculation of Chern numbers in hyperbolic band theory and verifying results numerically.

Contribution

It introduces a hyperbolic TKNN formula, interprets the Chern number topologically, and simplifies its calculation by focusing on Abelian states.

Findings

Hall conductivity quantized as -e^2C_{ij}/h

Chern number can be computed from Abelian states alone

Numerical verification in hyperbolic Haldane model

Abstract

We establish a connection between the electromagnetic Hall response and band topological invariants in hyperbolic Chern insulators by deriving a hyperbolic analog of the Thouless-Kohmoto-Nightingale-den Nijs (TKNN) formula. By generalizing the Kubo formula to hyperbolic lattices, we show that the Hall conductivity is quantized to $-e^2C_{ij}/h$, where $C_{ij}$ is the first Chern number. Through a flux-threading argument, we provide an interpretation of the Chern number as a topological invariant in hyperbolic band theory. We demonstrate that, although it receives contributions from both Abelian and non-Abelian Bloch states, the Chern number can be calculated solely from Abelian states, resulting in a tremendous simplification of the topological band theory. Finally, we verify our results numerically by computing various Chern numbers in the hyperbolic Haldane model.