黎曼幾何與物質光學性質
Riemannian Geometry of Resonant Optical Responses

Junyeong Ahn, Guang-Yu Guo*, Naoto Nagaosa & Ashvin Vishwanath
Nature Physics 18, 290 (2022)
DOI: https://doi.org/10.1038/s41567-021-01465-z

多年來量子態的幾何學是理解電子系統對靜態電磁場響應的基礎， 如量子霍爾效應源自於二維電子氣電子態的貝里曲率及拓撲不變性。最近我們通過將偶極矩躍遷矩陣元看作復向量空間的切向量，引進黎曼度量張量、連接和曲率，為材料光學響應構建了一套黎曼幾何理論，並應用該理論預言在材料中三階光伏霍爾效應是由黎曼曲率主導的，還提出驗證此預言的實驗。

The geometry of quantum states is a basis for understanding the response of electronic systems to electromagnetic fields, e.g, the quantum Hall effect. Here, we construct a theory of Riemannian geometry for  resonant optical processes by identifying dipole transition matrix elements as tangent vectors, showing that optical responses are manifestations of the Riemannian geometry of quantum states. We predict that third-order photovoltaic Hall effects are related to the Riemann curvature tensor, which can be verified  experimentally.