Pinch-point spectral singularity from the interference of topological loop states

The pinch point is a spectral discontinuity found in the neutron diffraction image of spin ice. A similar spectral singularity is commonly observed in a broad range of systems that have a close connection with flat bands. We focus on the electron flat band and its two topologically distinct classes of wave functions: the compact localized state (CLS) and the noncontractible loop state (NLS). We establish their simple mathematical relationship, showing that different Bloch NLSs can be derived as momentum derivatives of a Bloch CLS, depending on the approaching direction toward the singular point. This CLS-NLS correspondence helps visualize the pinch point as an interference pattern among NLSs through a ``polarizer,'' which encodes the information about the location of singular momentum and the experimental techniques like spin-polarized photoemission spectroscopy. It helps extract topological information knitted to microscopic electronic and magnetic structures.