Topological Number of Edge States

TL;DR

This paper explores topological charges of edge states in four-dimensional systems, revealing their relation to monopoles and introducing a non-Abelian analogue of the TKNN number, with implications for string theory and non-commutative geometry.

Contribution

It introduces the concept of topological charges for edge states in 4D systems, linking them to monopoles and extending topological invariants to non-Abelian cases.

Findings

Edge states can have topological charges characterized by monopoles.

A non-Abelian analogue of the TKNN number is defined via an SU(2) monopole.

Edge monopoles appear in non-commutative momentum space under magnetic fields.

Abstract

We show that the edge states of the four-dimensional class A system can have topological charges, which are characterized by Abelian/non-Abelian monopoles. The edge topological charges are a new feature of relations among theories with different dimensions. From this novel viewpoint, we provide a non-Abelian analogue of the TKNN number as an edge topological charge, which is defined by an SU(2) 't Hooft-Polyakov BPS monopole through an equivalence to Nahm construction. Furthermore, putting a constant magnetic field yields an edge monopole in a non-commutative momentum space, where D-brane methods in string theory facilitate study of edge fermions.