The bulk-edge correspondence for curved interfaces

TL;DR

This paper extends the bulk-edge correspondence theorem to curved interfaces in topological insulators, showing that edge conductance remains quantized and related to Hall conductance even with non-straight boundaries.

Contribution

It provides a rigorous proof that curved boundaries in topological insulators support quantized edge currents, generalizing previous results for straight boundaries.

Findings

Edge conductance is an integer multiple of Hall conductance for curved interfaces.

Supports the existence of edge currents regardless of boundary shape.

Provides a mathematical foundation for experimental observations of topological insulators.

Abstract

The bulk-edge correspondence is a condensed matter theorem that relates the conductance of a Hall insulator in a half-plane to that of its (straight) boundary. In this work, we extend this result to domains with curved boundaries. Under mild geometric assumptions, we prove that the edge conductance of a topological insulator sample is an integer multiple of its Hall conductance. This integer counts the algebraic number of times that the interface (suitably oriented) enters the measurement set. This result provides a rigorous proof of a well-known experimental observation: arbitrarily truncated topological insulators support edge currents, regardless of the shape of their boundary.