Riemannian Geometry Theory of Nonlinear Optical Properties of Materials

Bulk photovoltaic effect (BPVE) is a second-order nonlinear optical effect that generates dc photocurrents in a noncentrosymmetric solid under light irradiation. Topological semimetals are emerging as efficient infrared and terahertz photodetectors due to this promising mechanism. Recently, Professor Guo of NTU’s Department of Physics, together with coworkers from RIKEN and University of Tokyo, Japan as well as Harvard University, USA, investigated the second-order optical conductivity in topological semimetals described by Dirac and Weyl models [1]. Through symmetry and power counting analysis, they showed that Dirac and Weyl points with tilted cones exhibit the leading low-frequency divergence. They also uncovered the complete quantum geometric meaning of the low-frequency BPVE (Table I). For example, they found that the injection current is controlled by the quantum metric and Berry curvature, whereas the shift current is governed by the Christoffel symbols. They also performed material specific first-principles calculations, and found that gigantic photoconductivity at low frequencies is due to the divergent behavior of the geometric quantities near the Dirac and Weyl points. This work [1] brings new insights into the BPVE as well as the design of semimetal-based terahertz photodetectors.

Geometry of quantum states is a powerful concept for understanding responses of electronic systems to static electromagnetic fields, as exemplified by the quantum Hall effect. However, it has been challenging to relate quantum geometry with optical responses. The main obstacle is that optical transitions are properties of a pair of states, while existing geometrical properties are defined for a single state. Professor Guo and co-workers very recently constructed a Riemannian geometry theory for all-order optical processes, by identifying transition dipole moment matrix elements as tangent vectors (FIG. 1). They showed that optical responses can be generally thought of as manifestations of the Riemannian geometry of quantum states. They also applied their theory to show that third-order photovoltaic Hall effects are related to the Riemann curvature tensor and demonstrate an experimentally accessible regime where they dominate the response (Fig. 1). This intriguing discovery was recently published in top-notch physics journal Nature Physics [2].

[1] J. Ahn, G.-Y. Guo and N. Nagaosa, Physical Review X 10, 041041 (2020)
[2] J. Ahn, G.-Y. Guo, N. Nagaosa and A. Vishwanath, Nature Physics 18, 290 (2022)