Berry Curvature and the $Z_2$ Topological Invariants of Spin-Orbit-Coupled Bloch bands.

The (``anomalous'') integer quantum Hall effect can occur in non-interacting models of band insulators with broken time-reversal- ($T$-)symmetry where the sum of Chern invariants of occupied bands of Bloch states is non-zero. These topological invariants can be computed from the zeroes of certain functions in the Brillouin zone (BZ), but have a simpler formulation as BZ-integrals of Berry curvature. Recently, Kane and Mele found that $T$-invariant 2D systems with strong spin-orbit coupling possess a ``$Z_2$'' ($+$ or $-$) analog of the Chern invariant, which they formulated in terms of zero-counting arguments (3D generalizations have also been found). I give an alternate formulation in terms of Berry-curvature integrals, in the case that spatial-inversion- ($I$-)symmetry is broken, but $T$-symmetry is not. In 2D, such bands generically form a genus-5 2-manifold, with antipodal points paired by Kramers degeneracy: the $Z_2$ invariant is obtained by integration over a Kramers-distinct half-manifold; the 3D case is similar. I also discuss the case of doubly-degenerate bands with unbroken $I$-symmetry: despite recent suggestions, it does not appear that the $Z_2$ invariant of such systems can be obtained purely from knowledge of the parity quantum numbers at $T$-invariant points in the BZ.