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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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