Topological insulators on a Mobius Strip

TL;DR

This paper investigates topological insulators on a Mobius strip, revealing how edge current patterns can or cannot detect the strip's twist depending on the insulator type, challenging traditional topological invariants.

Contribution

It explores topological insulators on non-orientable surfaces, demonstrating limitations of edge current patterns in detecting topological features on a Mobius strip.

Findings

Edge current flow patterns can detect the Mobius twist for Chern insulators.

Edge current flow patterns cannot detect the twist for spin Hall insulators.

Traditional topological invariants are not defined on non-orientable surfaces.

Abstract

We study the two dimensional Chern insulator and spin Hall insulator on a non-orientable Riemann surface, the Mobius strip, where the usual bandstructure topological invariant is not defined. We show that while the flow pattern of edge currents can detect the twist of the Mobius strip in the case of Chern insulator, it can not do so in spin Hall insulator.