Extended Data Fig. 9: Schematic diagrams of spin configuration and hole density of an S₂ ligand in a Ni₂S₁₀ cluster.

a, Zhang–Rice triplet (ZRT) lattice; b, two Zhang–Rice singlet (ZRS) excited lattices with the zig-zag AFM order. Large (small) red and blue arrows refer to the spin direction of Ni 3d (S 3p) holes. Green ellipsoids indicate edge-shared S sites, in which spins of two ligand holes are antiparallel with one another. Magenta circles highlight sites of ZRS excitations. Open arrows inside magenta circles represent the direction of anticipated electric dipoles for the ZRS excitation. Owing to polarization reversal with the Ni spin flip, polarized PL along the b direction (the ‘armchair’ direction of the hexagonal Ni network) has destructive contributions from the opposite spin orientations, while the polarized PL along the a direction (the zig-zag direction) has not. This can explain the observed PL anisotropy shown in Extended Data Fig. 4. c–f, Hole density of an S₂ ligand as a function of the strength of the long-range magnetic field, h: c, e, for the ground state (GS) with J_p = 0 (c) and J_p = 0.4 eV (e); and d, f, for the excited state near 1.44 eV characterized by the ZRS state with J_p = 0 (d) and J_p = 0.4 eV (f). Other parameters are presented in Extended Data Table 1. In c–f, FM (AFM) indicates that applied long-range magnetic fields at two Ni sites are parallel (antiparallel). We assume the field direction is along the a axis because the result is almost same regardless of the field direction.