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 Z2 topological phases. We review several proposals for Z2 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 Z2 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.

Wannier functions and Z_2 invariants in time-reversal symmetric topological insulators / Cornean, Horia D; Monaco, Domenico; Teufel, Stefan. - In: REVIEWS IN MATHEMATICAL PHYSICS. - ISSN 0129-055X. - 29:(2017), pp. 1730001-1-1730001-66. [10.1142/S0129055X17300011]

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

Monaco, Domenico
;
Teufel, Stefan
2017

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 Z2 topological phases. We review several proposals for Z2 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 Z2 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.
2017
Bloch frames; fermionic time-reversal symmetry; topological insulators; Wannier functions; Z_2 invariants
01 Pubblicazione su rivista::01a Articolo in rivista
Wannier functions and Z_2 invariants in time-reversal symmetric topological insulators / Cornean, Horia D; Monaco, Domenico; Teufel, Stefan. - In: REVIEWS IN MATHEMATICAL PHYSICS. - ISSN 0129-055X. - 29:(2017), pp. 1730001-1-1730001-66. [10.1142/S0129055X17300011]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/1327319
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