Quantitative relationship between polarization differences and the zone-averaged shift photocurrent

A relationship is derived between differences in electric polarization between bands and the "shift vector" that controls part of a material's bulk photocurrent, then demonstrated in several models. Electric polarization has a quantized gauge ambiguity and is normally observed at surfaces via the surface charge density, while shift current is a bulk property and is described by shift vector gauge invariant at each point in momentum space. They are connected because the same optical transitions that are described in shift currents pick out a relative gauge between valence and conduction bands. We also discuss subtleties arising when there are points at the Brillouin zone where optical transitions are absent. We conclude that two-dimensional materials with significant interband polarization differences should have high bulk photocurrent, meaning that the modern theory of polarization can be used as a straightforward way to search for bulk photovoltaic material candidates.