${\mathbb{Z}}_2$ invariants of topological insulators as geometric obstructions

Abstract

We consider a gapped periodic quantum system with time-reversal symmetry of fermionic (or odd) type, i.e. the time-reversal operator squares to -1. We investigate the existence of periodic and time-reversal invariant Bloch frames in dimensions 2 and 3. In 2d, the obstruction to the existence of such a frame is shown to be encoded in a ${\mathbb{Z}}_2$-valued topological invariant, which can be computed by a simple algorithm. We prove that the latter agrees with the Fu-Kane index. In 3d, instead, four ${\mathbb{Z}}_2$ invariants emerge from the construction, again related to the Fu-Kane-Mele indices. When no topological obstruction is present, we provide a constructive algorithm yielding explicitly a periodic and time-reversal invariant Bloch frame. The result is formulated in an abstract setting, so that it applies both to discrete models and to continuous ones.

48 pages, 3 figures. Version 2: final version, to appear in CMP. A new section (Section 7), containing a proof of the completeness of the classification provided by the Z_2-invariants (Theorem 6), has been added. The behavior of the Z_2-indices with respect to a change of lattice basis, is now discussed at the end of Section 6.4

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• Symmetry and localization in periodic crystals: triviality of Bloch bundles with a fermionic time-reversal symmetry
• Wannier functions and Z_2 invariants in time-reversal symmetric topological insulators
• On the construction of composite Wannier functions
• A new approach to transport coefficients in the quantum spin Hall effect
• Gauge-theoretic invariants for topological insulators: A bridge between Berry, Wess-Zumino, and Fu-Kane-Mele
• Spin Conductance and Spin Conductivity in Topological Insulators: Analysis of Kubo-like terms
• Anyonic Topological Order in Twisted Equivariant Differential (TED) K-Theory
• Existence and computation of generalized Wannier functions for non-periodic systems in two dimensions and higher
• Parseval frames of exponentially localized magnetic Wannier functions
• The Iterated Projected Position Algorithm for Constructing Exponentially Localized Generalized Wannier Functions for Periodic and Non-Periodic Insulators in Two Dimensions and Higher
• On the construction of Wannier functions in topological insulators: the 3D case
• The FKMM-invariant in low dimension
• Approximate symmetries and conservation laws in topological insulators and associated $\mathbb{Z}$-invariants
• Numerical construction of Wannier functions through homotopy
• 2d Fu-Kane-Mele invariant as Wess-Zumino action of the sewing matrix
• Symmetric Fermi projections and Kitaev's table: topological phases of matter in low dimensions
• Coherent electronic transport in periodic crystals
• Differential geometric invariants for time-reversal symmetric Bloch-bundles II: The low dimensional "Quaternionic" case
• Topology vs localization in synthetic dimensions
• A ${\mathbb{Z}}_2$-Topological Index for Quasi-Free Fermions