Can harmonic spectra measure Berry curvature of a crystal?
In plain words
In crystals lacking mirror symmetry, electrons pushed by a field also drift sideways because of a geometric property of their bands called Berry curvature. Harmonics polarized at right angles to the laser might read this property out, if other effects can be excluded.
Precise statement
In noncentrosymmetric materials (monolayer MoS2, hBN, $\alpha$-quartz) the anomalous velocity $v = -(e/\hbar)\ E\ x\ \Omega(k)$, with $\Omega(k)$ the Berry curvature, generates even-order harmonics polarized perpendicular to the drive. Question: can the perpendicular even-harmonic amplitudes and phases be inverted to give $\Omega(k)$ along the driven trajectories with a stated accuracy, given interband contributions and propagation. An answer is a retrieval in a material with independently known Omega(k) that agrees within the stated uncertainty, or a proof that the inversion is not unique.
What would settle it
A blind retrieval of $\Omega(k)$ from harmonic data in a material whose Berry curvature is fixed by ab initio band theory and by an independent transport or optical measurement.
Status in the literature
Retrievals were reported in $\alpha$-quartz (Luu and Woerner, Nature Communications 2018) and even harmonics attributed to Berry curvature in MoS2 (2017); uniqueness and model dependence of the inversion remain debated.