Quantum State Geometry and Applications to Observables in Crystals

Geometrical and topological quantities can appear in optical properties and other aspects that go beyond the usual ground-state or adiabatic assumptions. Previous examples include the prediction of a universal scale and distinctive frequency dependence for the chiral photocurrent in Weyl semimetals in terms of material-independent fundamental constants, based on a quantization in the non-interacting limit, but such approaches were limited to specific materials and circumstances. This talk introduces a general theory of the multi-state wavefunction geometry that controls many optical properties [1,2], clarifying the existence of separate physical contributions within the overall curvature tensor. Part of the frequency-integrated optical response, and in many materials the dominant part, is determined by the skewness of ground-state orbitals through a sum rule that generalizes a well-known rule in linear optics. As an example, we use the theory to predict responses in some transition metal dichalcogenides. Its formulation in terms of explicitly gauge-invariant projection operators [2] makes practical computations considerably easier and suggests that related geometric objects may control other multi-state properties as well.

[1] Alexander Avdoshkin, Johannes Mitscherling, and Joel E. Moore, The multi-state geometry of shift current and polarization, Phys. Rev. Lett. 135, 066901 (2025).

[2] Johannes Mitscherling, Alexander Avdoshkin, and Joel E. Moore, Gauge-invariant projector calculus for quantum state geometry and applications to observables in crystals, Phys. Rev. B 112, 085104 (2025).