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 [Formula: see text] topological phases. We review several proposals for [Formula: see text] indices that distinguish these topological phases, including the ones by Fu–Kane [16], Prodan [33], Graf–Porta [24] and Fiorenza–Monaco–Panati [27]. We show that all these formulations are equivalent. In particular, this allows to prove a geometric formula for the [Formula: see text] invariant of 2-dimensional TRS topological insulators, originally indicated in [16], which expresses it in terms of the Berry connection and the Berry curvature.
Maximally localized generalized Wannier functions for composite energy bands
