Connecting second-order optical response with the band structure geometry of Weyl semimetals*

Nonlinear optical response is well studied in the context of semiconductors and has gained a renaissance in studies of topological materials in the recent decade. In non-magnetic materials the response is believed to root in the Berry curvature of the material band structure. Here, we revisit the general formalism for the second-order optical response with finite lifetimes, focusing on the consequences of the time-reversal-symmetry breaking. We identify three physical mechanisms to generate a dc photocurrent, i.e. the Berry curvature, the quantum metric and the diabatic motion. All three effects are non-zero when time-reversal symmetry is broken. They can be understood intuitively from the anomalous acceleration; the first two terms are respectively the antisymmetric and symmetric parts of the quantum geometric tensor. The last term is due to the dynamical antilocalization that refers to the phase accumulation between time-reversed fermion loops.