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

The shift current is the intrinsic bulk photovoltaic response of a crystal without an inversion center. In graphene multilayers, recent calculations report large band-edge shift currents that reverse sign under a gate or displacement field, a behavior that neither the quantum metric nor the Berry curvature captures. We show that this behavior follows from a symmetry principle: at the absorption edge of an inversion-broken, two-dimensional gapped Dirac-like system, an emergent low-energy rotational symmetry forbids the response, and the trigonal warping, which reduces this emergent symmetry to the lattice's three-fold rotation, switches the current on linearly in its strength. We formulate this as an exact angular selection rule that unifies the band-edge responses of multilayer graphene and of the kagome lattice. In the multiband structures, the released current is governed by a signed, detuning-weighted three-point Bargmann invariant of the Bloch states: the optical amplitude remains fixed while the sign alignment of the Bargmann triangles grows with the warping, a phase-coherence effect captured by a bounded coherence factor and invisible to any positive-definite figure of merit, the quantum metric included. In bilayer and trilayer graphene, a gate voltage alone drives the sign reversal, making the band-edge shift current a parameter-free, gate-switchable bulk photovoltaic response.