Topological surface states in three-dimensional magnetic insulators

TL;DR

This paper explores topologically nontrivial surface states in three-dimensional magnetic insulators, introducing a new invariant called the Hopf invariant, and provides computational tools and models to analyze these states.

Contribution

It introduces an efficient algorithm for computing the Hopf invariant in magnetic band structures and presents a tight-binding model realizing nontrivial Hopf insulators.

Findings

The Hopf invariant can classify topological phases in magnetic insulators.

A double-exchange-like tight-binding model exhibits nontrivial Hopf topology.

Numerical analysis reveals the properties of surface states in Hopf insulators.

Abstract

An electron moving in a magnetically ordered background feels an effective magnetic field that can be both stronger and more rapidly varying than typical externally applied fields. One consequence is that insulating magnetic materials in three dimensions can have topologically nontrivial properties of the effective band structure. For the simplest case of two bands, these "Hopf insulators" are characterized by a topological invariant as in quantum Hall states and Z_2 topological insulators, but instead of a Chern number or parity, the underlying invariant is the Hopf invariant that classifies maps from the 3-sphere to the 2-sphere. This paper gives an efficient algorithm to compute whether a given magnetic band structure has nontrivial Hopf invariant, a double-exchange-like tight-binding model that realizes the nontrivial case, and a numerical study of the surface states of this model.