Topological linear response of hyperbolic Chern insulators

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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mo>−</mml:mo> <mml:msup> <mml:mi>e</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:msub> <mml:mi>C</mml:mi> <mml:mrow> <mml:mi>i</mml:mi> <mml:mi>j</mml:mi> </mml:mrow> </mml:msub> <mml:mi>/</mml:mi> <mml:mi>h</mml:mi> </mml:mrow> </mml:math> , where C_{ij} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi>C</mml:mi> <mml:mrow> <mml:mi>i</mml:mi> <mml:mi>j</mml:mi> </mml:mrow> </mml:msub> </mml:math> 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.

Topics & Concepts