  We develop a framework for computing light dark matter direct detection rates
through single phonon and magnon excitations via general effective operators.
Our work generalizes previous calculations focused on spin-independent
interactions involving the total nucleon and electron numbers $N$ (the usual
route to excite phonons) and spin-dependent interactions involving the total
electron spin $S$ (the usual route to excite magnons), leading us to identify
new responses involving the orbital angular momenta $L$, as well as spin-orbit
couplings $L\otimes S$ in the target. All four types of responses can excite
phonons, while couplings to electron's $S$ and $L$ can also excite magnons. We
apply the effective field theory approach to a set of well-motivated
relativistic benchmark models, including (pseudo-)scalar mediated interactions,
and models where dark matter interacts via a multipole moment, such as a dark
electric dipole, magnetic dipole or anapole moment. We find that couplings to
point-like degrees of freedom $N$ and $S$ often dominate dark matter detection
rates, implying that exotic materials with orbital $L$ order or large
spin-orbit couplings $L\otimes S$ are not necessary to have strong reach to a
broad class of DM models. We highlight that phonon based crystal experiments in
active R&D (such as SPICE) will probe light dark matter models well beyond
those having a simple spin-independent interaction, including e.g. models with
dipole and anapole interactions. Lastly, we make publicly available a code,
PhonoDark, which computes single phonon production rates in a wide variety of
materials with the effective field theory framework.
