Symmetry and localization in periodic crystals: triviality of Bloch bundles with a fermionic time-reversal symmetry

TL;DR

This paper uses group- and bundle-theoretic methods to analyze how symmetries, especially time-reversal, influence electron localization and topological properties in crystalline solids, demonstrating triviality of Bloch bundles under certain symmetries.

Contribution

It shows how time-reversal symmetry ensures the triviality of Bloch bundles and introduces new topological invariants related to topological insulators.

Findings

Time-reversal symmetry implies Bloch bundle triviality.

Almost-exponential localization linked to smooth Bloch frames.

New topological invariants consistent with Fu, Kane, and Mele's indices.

Abstract

We describe some applications of group- and bundle-theoretic methods in solid state physics, showing how symmetries lead to a proof of the localization of electrons in gapped crystalline solids, as e.g. insulators and semiconductors. We shortly review the Bloch-Floquet decomposition of periodic operators, and the related concepts of Bloch frames and composite Wannier functions. We show that the latter are almost-exponentially localized if and only if there exists a smooth periodic Bloch frame, and that the obstruction to the latter condition is the triviality of a Hermitian vector bundle, called the Bloch bundle. The role of additional $\mathbb{Z}_2$-symmetries, as time-reversal and space-reflection symmetry, is discussed, showing how time-reversal symmetry implies the triviality of the Bloch bundle, both in the bosonic and in the fermionic case. Moreover, the same $\mathbb{Z}_2$-symmetry allows to define a finer notion of isomorphism and, consequently, to define new topological invariants, which agree with the indices introduced by Fu, Kane and Mele in the context of topological insulators.