Symmetry selection rule for the band-edge shift current in two dimensions

Abstract

The shift current is the intrinsic bulk photovoltaic response of a crystal without an inversion center. In graphene multilayers, calculations report large band-edge shift currents that reverse sign under a gate, a behavior that neither the quantum metric nor the Berry curvature captures. We show that this follows from a symmetry principle. At the absorption edge of a gapped, inversion-broken two-dimensional Dirac-like system, an emergent rotational symmetry forbids the response, and the trigonal warping switches it on linearly in its strength, an angular selection rule that unifies multilayer graphene and the kagome lattice. The released current is governed by a signed, detuning-weighted three-point Bargmann invariant: the optical amplitude stays fixed while the sign alignment of the Bargmann triangles grows with the warping, invisible to any positive-definite figure of merit. In bilayer and trilayer graphene the gate alone reverses the sign, making the band-edge shift current a gate-switchable bulk photovoltaic response.