Nonlinear photoconductivities and quantum geometry of chiral multifold fermions

Chiral multifold fermions are quasi-particles that appear only in chiral crystals such as transition metal silicides in the cubic B20 structure (i.e., the CoSi family), and they may show exotic physical properties. Here we study the injection and shift photoconductivities and also the related geometrical quantities for several types of chiral multifold fermions, including spin-1/2 as well as pseudospin-1 and -3/2 fermions, dubbed as Kramers Weyl, triple point and Rarita-Schwinger-Weyl (RSW) fermions, respectively. We utilize the minimal symmorphic model to describe the triple point fermions (TPF). We also consider the more realistic model Hamiltonian for the CoSi family including both linear and quadratic terms. We find that circular injection currents are quantized as a result of the Chern numbers carried by the multifold fermions within the linear models. Surprisingly, we discover that in the TPF model, linear shift conductivities are proportional to the pseudo spin-orbit coupling and independent of photon frequency. In contrast, for the RSW and Kramer Weyl fermions, the linear shift conductivity is linearly proportional to photon frequency. The numerical results agree with the power-counting analysis for quadratic Hamiltonians. The frequency independence of the linear shift conductivity could be attributed to the strong resonant symplectic Christoffel symbols of the flat bands. Moreover, the calculated symplectic Christoffel symbols show significant peaks at the nodes, suggesting that the shift currents are due to the strong geometrical response near the topological nodes.