Abstract
In applied science, nonlinear optical (NLO) responses of materials especially second-harmonic generation [1] and bulk photovoltaic effect (BPVE) [2], have received renewed interests in recent years perhaps because of their promising applications in next generation opto-electronics, photovoltaic solar cells and high sensitive THz radiation detection. On the other hand, in condensed matter theory and mathematical physics, understanding responses of materials to static electromagnetic fields, e.g., quantum Hall effect [3] and anomalous Hall conductivity [4], in terms of geometry of quantum states has become a central topic in recent decades. In this talk, I will present the main findings of our recent endeavor to understand NLO properties of solids in terms of geometric quantities of the manifold of electronic energy bands in the solids. In particular, we found that in BPVE, the injection current is controlled by the quantum metric and Berry curvature while the shift current is governed by the Christoffel symbols near the nodal points in topological semimetals [5]. More recently, through both analytical and density-functional theory calculations, we found that resonant optical responses of all orders could be casted in the framework of Riemannian geometry [6]. Because I started working on BPVE and quantum geometry only early last year when I was on my sabbatical visit to RIKEN, Japan, the main purpose of this talk is to hear the comments/corrections from the audience on the intriguing but not-matured ideas.