Direct measurement of the quantum metric tensor in solids

The quantum metric tensor is a central geometric quantity in modern physics that determines the distance between nearby quantum states. Despite numerous studies highlighting its relevance to fundamental physical phenomena in solids, the method for measuring the complete quantum metric tensors in real solid-state materials remains unknown. Here, we report the first direct measurement of the full quantum metric tensors of Bloch electrons in solids using black phosphorus as a representative material. The key idea is to extract the momentum space distribution of the pseudospin texture of the valence band from the polarization dependence of angle-resolved photoemission spectroscopy measurement. The spectroscopic probe of all possible quantum metric tensor components is poised to significantly advance our understanding of quantum geometric responses in a wide class of crystalline systems., Theoretical data is collected from the density functional theory calculations and electronic band calculations based on the tight-binding model. Also, the experimental data is collected from the angle-resolved photoemission spectroscopy., , # Direct measurement of the quantum metric tensor in solids

[https://doi.org/10.5061/dryad.jwstqjqmq](https://doi.org/10.5061/dryad.jwstqjqmq)

We use three methods to collect the data:

1. We calculate the eigenstate and the eigenvalues of the model Hamiltonian based on the tight-binding model. For this calculation, we use Mathematica. This method is used to calculate the data in the folders Fig1, Fig3, Fig4, FigS1, FigS2, FigS11, and FigS12.
2. We directly calculate the electronic properties of the black phosphorus through the density functional theory calculations. This method is used to calculate the data in the folders Fig3, Fig4, FigS7, FigS8, FigS9, and FigS10.
3. We perform the angle-resolved photoemission spectroscopy (ARPES) for bulk black phosphorus and measure the photoemitted electron intensity. This data is in the folders Fig2, FigS4, FigS5, and FigS6.

## Description of the data and file structure

Here, we explain how the data in each folder is calculated or measured.

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