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\(\mathbb Z_2\) invariants of topological insulators as geometric obstructions. (English) Zbl 1346.81158
The physical intuition behind the present research is that of a new class of materials, called nowadays time-reversal symmetric topological insulators. Their topological quantum phases are labeled by integers modulo \(2\). The Fu-Kane analysis of the exemplary tight-binding model governing the behavior of electrons on a honeycomb lattice, subject to nearest neighbor interactions, resulted in the identification of the \(\mathbb{Z}_2\) index to label the topological phases of \(2d\) TRS topological insulators. L. Fu and C. L. Kane [“Time reversal polarization and a \(Z_2\) adiabatic spin pump”, Phys. Rev. B 74, No. 19 Article ID 195312, 13 p. (2006; doi:10.1103/PhysRevB.74.195312)] argued that this index measures the obstruction to the existence of a continuous periodic Bloch frame, which is compatible with the time-reversal symmetry. Analogous proposals have been formulated in \(3d\) systems by L. Fu, et al. [“Topological insulators in three dimensions”, Phys. Rev. Lett. 98, No. 10, Article ID 106803, 4 p. (2007; doi:10.1103/PhysRevLett.98.106803)]. In the present paper the above claims are given a mathematically rigorous status. A geometric characterization is given of \(\mathbb{Z}_2\) indices as topological obstructions to the existence of continuous-periodic and time-reversal symmetric Bloch frames. It is proven that the Fu-Kane index actually is the topological invariant. In \(3d\) one arrives at four \(\mathbb{Z}_2\)-valued topological obstructions that give an unambiguous status to the Fu-Kane-Mele indices. When there is no topological obstruction, an explicit algorithm has been proposed to construct global smooth Bloch frames which are periodic and time-reversal symmetric. The main advantage of the method is that it is based on fundamental symmetries of the system which makes the approach model-independent.

MSC:
81V70 Many-body theory; quantum Hall effect
46A63 Topological invariants ((DN), (\(\Omega\)), etc.) for locally convex spaces
55S25 \(K\)-theory operations and generalized cohomology operations in algebraic topology
19L10 Riemann-Roch theorems, Chern characters
30D45 Normal functions of one complex variable, normal families
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