Non-linear transport in multifold semimetals

Transport measurements are a powerful way to probe the electronic structure of quantum materials, but the information they contain is often convoluted. Yet, in particular for simple low-energy fermiologies, and by combining linear and non-linear responses, definite conclusion can be drawn -- such as, for instance, in the case of the circular photogalvanic effect in Weyl semimetals. Here, we derive the complete DC intrinsic transport response functions up to third order in the applied electric field within Boltzmann theory that hold combined information about quantum geometry and band dispersion. We discuss the responses for multifold fermions at high-symmetry momenta in time-reversal symmetric crystals as well as their reduction by symmetry constraints. We exemplify in detail the cases of space group 213 and space group 199, which realize different multifold fermions, and show under which conditions these low-energy excitations can be differentially addressed through their bulk nonlinear responses, enabling nonlinear valley-tronics.