Wannier functions and Z_2 invariants in time-reversal symmetric topological insulators

TL;DR

This paper constructs exponentially localized Wannier functions for 1D and 2D time-reversal symmetric topological insulators, explores topological obstructions related to $$ invariants, and proves their equivalence and a geometric formula involving Berry phases.

Contribution

It provides a model-independent construction of Wannier functions in TRS topological insulators and establishes the equivalence of various $$ index formulations, including a geometric Berry connection-based formula.

Findings

Constructed Wannier functions in TRS topological insulators.

Proved equivalence of different $$ index definitions.

Derived a geometric formula for the $$ invariant using Berry connection and curvature.

Abstract

We provide a constructive proof of exponentially localized Wannier functions and related Bloch frames in 1- and 2-dimensional time-reversal symmetric (TRS) topological insulators. The construction is formulated in terms of periodic TRS families of projectors (corresponding, in applications, to the eigenprojectors on an arbitrary number of relevant energy bands), and is thus model-independent. The possibility to enforce also a TRS constraint on the frame is investigated. This leads to a topological obstruction in dimension 2, related to $\mathbb{Z}_2$ topological phases. We review several proposals for $\mathbb{Z}_2$ indices that distinguish these topological phases, including the ones by Fu--Kane [Phys. Rev. B 74 (2006), 195312], Prodan [Phys. Rev. B 83 (2011), 235115], Graf--Porta [Commun. Math. Phys. 324 (2013), 851] and Fiorenza--Monaco--Panati [Commun. Math. Phys., in press]. We show that all these formulations are equivalent. In particular, this allows to prove a geometric formula for the the $\mathbb{Z}_2$ invariant of 2-dimensional TRS topological insulators, originally indicated in [Phys. Rev. B 74 (2006), 195312], which expresses it in terms of the Berry connection and the Berry curvature.