$Z_2$ topology of bismuth

TL;DR

This paper uses advanced quasiparticle self-consistent $GW$ calculations to clarify the topological nature of bismuth, resolving discrepancies between previous theoretical predictions of trivial topology and experimental indications of non-trivial surface states.

Contribution

The study demonstrates that bismuth's surface states are topologically trivial when analyzed with quasiparticle self-consistent $GW$ calculations, resolving prior conflicting interpretations.

Findings

Surface states are consistent with trivial $Z_2$ invariant.

Extreme penetration depth affects surface state interpretation.

Bulk-like behavior requires films thicker than 400 nm.

Abstract

While first-principles calculations with different levels of sophistication predict a topologically trivial $Z_2$ state for bulk bismuth, some photoemission experiments show surface states consistent with the interpretation of bismuth being in a topologically non-trivial $Z_2$ state. We resolve this contradiction between theory and experiment by showing, based on quasiparticle self-consistent $GW$ calculations, that the experimental surface states interpreted as supporting a non-trivial phase are actually consistent with a trivial $Z_2$ invariant. We identify this contradiction as the result of a crosstalk effect arising from the extreme penetration depth of the surface states into the bulk of Bi. A film of Bi can be considered bulk-like only for thicknesses of about 1000 bilayers ($\approx$ 400 nm) and more.