Discrete Quantum Geometry for Nonlinear Optical Responses

For the two past decades, topology has become the centerpiece of many fields in condensed matter physics. In particular, k-space topology has led to many discoveries including topological insulators, Chern insulators, Weyl semimetals, and topological superconductors. These systems are predicted to exhibit unique electronic transport responses described by topological invariants evaluated through an integral of the Berry curvature among occupied states within the Fermi sea. Numerically, these calculations can prove challenging as certain electronic band features, such as band crossings, can lead to singularities, especially when close to the Fermi surface. An alternative approach however is to describe the responses using geometric properties of the Fermi surface. For example, by constructing a 3-dimensional discrete quantum manifold of the Fermi surface, it has recently been demonstrated that the intrinsic Hall conductivity resolved in spin can be precisely computed [1].

In this study, we generalize this geometric approach to nonlinear optical responses for gapless topological materials. We start with the circular photogalvanic effect (CPGE), which is predicted to directly measure the topological charge of three-dimensional metallic gapless systems using a topological property of the Fermi surface in the low frequency regime [2]. The numerical approach is demonstrated by computing the CPGE tensor for a time-reversal symmetric, but mirror and inversion broken multi-band Weyl semimetal. We then use this numerical approach to predict nonlinear optical responses for known noncentrosymmetric, metallic systems.

1. Jie-Xiang Yu et. al, Phys. Rev. B 104, 184408 (2021)

2. A. M. Rappe et. al, Phys. Rev. Res 3, L042032 (2021)