Topological fine structure of an energy band

TL;DR

This paper reveals that trivial energy bands can exhibit topologically nontrivial behavior under weak disorder, with extended states carrying opposite Chern numbers, explained by a new localizer index that uncovers a fine topological structure.

Contribution

It introduces the localizer index as a tool to detect topological features in trivial bands, showing they can host extended states with opposite Chern numbers due to disorder.

Findings

Trivial bands can host extended states with opposite Chern numbers.

The localizer index reveals a fine topological structure within trivial bands.

Weak disorder induces nontrivial extended states in trivial bands.

Abstract

A band with a nonzero Chern number cannot be fully localized by weak disorder. There must remain at least one extended state, which ``carries the Chern number.'' Here we show that a trivial band can behave in a similar way. Instead of fully localizing, arbitrarily weak disorder leads to the emergence of two sets of extended states, positioned at two different energy intervals, which carry opposite Chern numbers. Thus, a single trivial band can show the same behavior as two separate Chern bands. We show that this property is predicted by a topological invariant called a ``localizer index.'' Even though the band as a whole is trivial as far as the Chern number is concerned, the localizer index allows access to a topological fine structure. This index changes as a function of energy within the bandwidth of the trivial band, causing nontrivial extended states to appear as soon as disorder is introduced. Our work points to a previously overlooked manifestation of topology, which impacts the response of systems to impurities beyond the information included in conventional topological invariants.