Coherence from interference: a solvable model of sub-GeV dark matter-nucleus scattering

How do dark matter-nucleus interactions transition from the regimes of coherent scattering, where single phonons are produced, to that of individual nuclear recoils? Answering this question relies on understanding multiphonon excitations. Multiphonons are important for interpreting low-threshold direct detection experiments, yet are computationally prohibitive to compute. In this paper, we employ a 1D $N$-site crystal lattice model where dark matter scattering can be computed exactly. We show that the only difference between coherent and incoherent scattering is that conservation of crystal momentum is enforced in coherent scattering. The momentum conservation constraint becomes less important as more phonons are produced, yielding the transition to incoherent scattering. Using numerical calculations of the 1D structure factor, we also obtain quantitative validation of using an incoherent approximation to compute sub-GeV dark matter scattering in realistic 3D crystals.