Spin-orbit spillage as a measure of band inversion in insulators

We propose a straightforward and effective approach for quantifying the band inversion induced by spin-orbit coupling in band insulators. In this approach we define a quantity as a function of wave vector in the Brillouin zone (BZ) measuring the mismatch, or spillage, between the occupied states of a system with and without SOC. Plots of the spillage throughout the BZ provide a ready indication of the number and location of band inversions driven by SOC. To illustrate the method, we apply this approach to the two-band Dirac model, the 2D Kane-Mele model, a 2D Bi bilayer with applied Zeeman field, and to first-principles calculations of some 3D materials including both trivial and Z(2) topological insulators. We argue that the distribution of spillage in the BZ is closely related to the topological indices in these systems. Our approach provides a fresh perspective for understanding topological character in band theory, and should be helpful in searching for new materials with nontrivial band topology.