Half moons are pinch points with dispersion

Frustrated magnetism provides a fascinating playground for realizing exotic states of matter, such as the spin liquid. A characteristic signature for a U(1) spin liquid is the ``pinch point'', a distinctive singular pattern observed in neutron scattering experiments. Another pattern of an anisotropic ring, dubbed the ``half moon'', often accompanies the pinch point. Here, the authors reveal the underlying physics linking the two signatures. The half moons are essentially pinch points inscribed on a dispersing band. By analyzing the kagome lattice--Heisenberg antiferromagnet, this work presents a detailed study of how a unifying field theory accounts for both features intuitively.