Computing topological invariants without inversion symmetry

We consider the problem of calculating the weak and strong topological indices in noncentrosymmetric time-reversal ($T$) invariant insulators. In 2D we use a gauge corresponding to hybrid Wannier functions that are maximally localized in one dimension. Although this gauge is not smoothly defined on the two-torus,\footnote{A. A. Soluyanov and D. Vanderbilt, arXiv:1009.1415} it respects the $T$ symmetry of the system and allows for a definition of the ${Z}_2$ invariant in terms of time-reversal polarization.\footnote{L. Fu and C. L. Kane, Phys. Rev. B {\bf 74}, 195312 (2006)} In 3D we apply the 2D approach to $T$-invariant planes. We illustrate the method with first-principles calculations on GeTe and HgTe under $[100]$ and $[111]$ strain. Our approach is different from the one suggested previously by Fukui and Hatsugai\footnote{T. Fukui and Y. Hatsugai, J. Phys. Soc. Jpn. {\bf 76}, 053702 (2007)} and should be easier to implement in {\it ab initio} code packages. Time permitting, we will also discuss methods for decomposing the band space into $T$-paired Chern subspaces, and for carrying out a general construction of a Wannier representation for ${Z}_2$ insulators.