Nonlinear photoconductivities and quantum geometry of multifold fermions*

The injection and shift conductivities and their relevant geometrical quantities for multifold fermions, including pseudospin-1 and -3/2 excitations, are studied. For pseudospin-1 fermion, or the triple point fermion (TPF), we utilize the minimal symmorphic model which hosts TPFs protected by C4 rotation symmetry and an anticommuting mirror symmetry. The circular injection conductivity shows quantization as a result of the Chern number of the node. The linear shift conductivity of the TPF model is proportional to the pseudo spin-orbit coupling and independent of photon frequency.  For the pseudospin-3/2 excitation, we utilize the effective Hamiltonian for the CoSi family. The circular injection conductivity shows two consecutive plateaus as a result of the Chern number carried by each of the occupied bands. The linear shift conductivity is linearly dependent on the photon frequency. The corresponding Christoffel symbols show a significant peak near the node, suggesting the strong geodesic response in the momentum space near the node.