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Smooth gauge for topological insulators
We developed a technique for constructing smooth Bloch functions
for Z2 quantum spin Hall insulators. As the
initial step the occupied subspace of the system is decomposed
into a direct sum of two topologically non-trivial
subspaces with well defined Chern numbers - Chern bands.
This decomposition is different from the ones suggested
elsewhere and remains robust independently of underlying symmetries and features of
a particular model.
Starting with thus obtained Chern bands, we construct a
topologically
non-trivial unitary transformation that rotates the
occupied subspace into a direct sum of topologically
trivial subspaces.
Computing topological invariants without inversion symmetry
We considered the problem of calculating the weak and strong topological indices in noncentrosymmetric time-reversal invariant insulators. In 2D we used a gauge
corresponding to hybrid Wannier functions that are maximally localized in one dimension. Although this gauge is not smoothly defined on the Brillouin zone two-torus, it
respects the time-reversal symmetry of the system and allows for a definition of the Z2 invariant in terms of time-reversal polarization. In 3D we applied the 2D approach to
time-reversal invariant planes. The method was illustrated with first-principles calculations on GeTe and on HgTe under [001] and [111] strain.
Wannier representation of topological insulators
The problem of constructing Wannier functions for Z2 topological insulators in two dimensions was considered. It was well known that there is a topological obstruction to
the construction of Wannier functions for Chern insulators, but it had been unclear whether this is also true for the Z2 case. We considered the Kane-Mele tight-binding model,
which exhibits both normal (Z2-even) and topological (Z2-odd) phases as a function of the model parameters. In the Z2-even phase, the usual projection-based scheme can be
used to build the Wannier representation. In the Z2-odd phase, we did find a topological obstruction, but only if one insists on choosing a gauge that respects the time-reversal
symmetry, corresponding to Wannier functions that come in time-reversal pairs. If instead we are willing to violate this gauge condition, a Wannier representation becomes
possible. We presented an explicit construction of Wannier functions for the Z2-odd phase of the Kane-Mele model via a modified projection scheme followed by maximal
localization, and confirmed that these Wannier functions correctly represent the electric polarization and other electronic properties of the insulator.
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