We investigate the emergence and transformation of pinch-point singularities in the excitation spectrum of electronic flat-band systems on kagome and pyrochlore lattices with spin-orbit coupling (SOC) and Coulomb interactions. While pinch points are widely recognized as signatures of classical spin liquids, they also appear in electronic flat-band systems when there is a singular band-touching point to dispersive bands. We explore how SOC modifies the pinch-point structure in the chiral spin flat-band metallic state, which we term spinor-ice. The pinch-point profile can rotate or redistribute its spectral weight, governed by a prefactor in the spectral function that primarily depends on the direction of the ground-state spin polarization. We show that the spinor-ice state could be experimentally probed by rotating the spin polarization of the injected electron to infer internal magnetic structures. These observations are discussed in conjunction with angle-resolved photoemission spectroscopy and the application to the potential SOC flat-band material ${\mathrm{CsW}}_2{\mathrm{O}}_6$. We also demonstrate the persistent residual pinch-point features under Coulomb interactions and deviations from the ideal flat-band limit.

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• Charge order
• Flat bands
• Phase diagrams
• Spin-orbit coupling
• Kagome lattice
• Pyrochlores
• Extended Hubbard model
• Mean field theory
• Tight-binding model