Extended Data Fig. 7 | Calculated electronic structures of the pristine and defective (with a S vacancy) S-terminated surfaces. Band structures of the pristine (a) and the defective (b) S surface (in a 4 × 4 supercell). By comparing (a) and (b), a bound state is clearly visible as marked by the blue solid line and labelled by the blue arrow in (b). c, Spatial distribution of the wavefunction norm of the bound state marked in (b), which confirms its bound state nature. An isosurface of 3.0 × 10-4 e/Bohr3 is used. d, Spin-resolved DOS projected on the S vacancy, showing appreciable spin-polarization, in agreement with the experimental observation. We want to point out that the bound state is almost two-fold degenerated, while there are at least three bound states observed experimentally. The experiment-theory discrepancy reflects the limitation of standard DFT to explain the defect excitations in magnetic Weyl semimetals. The magnetic moment integrated within the sphere of radius 1.164 Å (the Wigner-Seitz radius of S atom) is 0.04 μB. The experimental data (Fig. 2d-h) indicate that the size of the localized polaron is much larger and contains a cluster of atoms around the S vacancy. Therefore we estimate the contribution to the magnetic moment of the localized polaron from one S vacancy and three neighboring Co atoms. The total
pseudospin (electron spin + atomic orbital) magnetic moment from the calculation is 0.85 μB. For the three Co atoms nearest to S vacancy, the orbital moment is 0.09 μB and the total magnetic moment is 0.81 μB. In the standard formula of DFT for non-collinear spin calculations, the diamagnetic term of the Dirac equation is ignored, which cannot account for the diamagnetic orbital magnetization contribution. This paramagnetic spin-orbit moment follows the magnetic field direction and cannot explain the observed net diamagnetic moment of 1.35 μB that always points in the opposite direction of the magnetic field. The exact diamagnetic orbital moment thus equals to the value of the net diamagnetic moment plus that of the paramagnetic moment revealed by DFT, i.e. being of the order ~ (1.35 + 0.85) =2.20 μB. The discrepancy on diamagnetic moment between our experiment and DFT calculation, indeed, supports the spin-orbit nature of the found polaron.