English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We provide a constructive proof of exponentially localized Wannier functionsand related Bloch frames in 1- and 2-dimensional time-reversal symmetric (TRS)topological insulators. The construction is formulated in terms of periodic TRSfamilies of projectors (corresponding, in applications, to the eigenprojectorson an arbitrary number of relevant energy bands), and is thusmodel-independent. The possibility to enforce also a TRS constraint on theframe 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 thesetopological 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., inpress]. We show that all these formulations are equivalent. In particular, thisallows to prove a geometric formula for the the $\mathbb{Z}_2$ invariant of2-dimensional TRS topological insulators, originally indicated in [Phys. Rev. B74 (2006), 195312], which expresses it in terms of the Berry connection and theBerry curvature.

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