Domenico Monaco

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

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

Chern and Fu–Kane–Mele Invariants as Topological Obstructions

\mathbb{Z}_2

Gauge-theoretic invariants for topological insulators: a bridge between Berry, Wess–Zumino, and Fu–Kane–Mele

topological phases. We review several proposals for

Optimal Decay of Wannier functions in Chern and Quantum Hall Insulators

\mathbb{Z}_2

On the Construction of Wannier Functions in Topological Insulators: the 3D Case

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

Adiabatic currents for interacting electrons on a lattice

\mathbb{Z}_2

Parseval frames of localized Wannier functions

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.

The Localization Dichotomy for gapped periodic quantum systems

The use of topological invariants to describe geometric phases of quantum matter has become an essential tool in modern solid state physics. The first instance of this paradigmatic trend can be traced to the study of the quantum Hall effect, in which the Chern number underlies the quantization of the transverse Hall conductivity. More recently, in the framework of time-reversal symmetric topological insulators and quantum spin Hall systems, a new topological classification has been proposed by Fu, Kane and Mele, where the label takes value in \(\mathcal{Z}_{2}\).

Parseval frames of exponentially localized magnetic Wannier functions.

We establish a connection between two recently proposed approaches to the understanding of the geometric origin of the Fu–Kane–Mele invariant