Half moons are pinch points with dispersion

"Pinch points," singular features observed in (quasi-)elastic neutron scattering, are a widely discussed hallmark of spin liquids with an emergent gauge symmetry. Much less attention has been paid to "half-moons," distinctive crescent patterns at finite energy, which have been observed in experiments on a number of pyrochlore magnets, and in a wide range of model calculations. Here we unify these two phenomena within a single framework, paying particular attention to the case of ordered, or field-saturated states, where pinch points and half moons can be found in bands of excitations above a gap. We find that half moons are nothing other than pinch points inscribed on a dispersing band. Molecular dynamics simulations of the kagome-lattice antiferromagnet are used to explore how these bands evolve into the ground state and excitations of a classical spin liquid. We explicitly demonstrate that this theory can reproduce the pinch points and half moons observed in Nd₂Zr₂O₇.