We study disordered topological insulators with time reversal symmetry. Relying on the noncommutative index theorem which relates the Chern number to the projection onto the Fermi sea and the magnetic flux operator, we give a precise definition of the ℤ2 index which is a noncommutative analogue of the Atiyah-Singer ℤ2 index. We prove that the noncommutative ℤ2 index is robust against any time reversal symmetric perturbation including disorder potentials as long as the spectral gap at the Fermi level does not close.
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Note that an arbitrary two-dimensional tight-binding model can be mapped onto the model on ℤ2 with suitably chosen hopping integrals.
Restricting the region to one-half of the Brillouin zone is essential to this argument, because one cannot find a nonsingular connection on the whole Brillouin zone.