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

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.

54 pages, 5 figures. Minor changes with respect to v1, including an updated bibliography. This version is published in Rev. Math. Phys

References in corpus (5)

• Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells
• Topological Crystalline Insulators
• Time Reversal Polarization and a Z_2 Adiabatic Spin Pump
• Homotopy Theory of Strong and Weak Topological Insulators
• Manifestly gauge independent formulations of the Z2 invariants

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