  Circularly polarized lattice vibrations carry angular momentum and lead to
magnetic responses in applied magnetic fields or when resonantly driven with
ultrashort laser pulses. Recent measurements have found responses that are
orders of magnitude larger than those calculated in prior theoretical studies.
Here, we present a microscopic model for the effective magnetic moments of
chiral phonons in magnetic materials that is able to reproduce the
experimentally measured magnitudes and that allows us to make quantitative
predictions for materials with giant magnetic responses using microscopic
parameters. Our model is based on orbit-lattice couplings that hybridize
optical phonons with orbital electronic transitions. We apply our model to two
types of materials: $4f$ rare-earth halide paramagnets and $3d$
transition-metal oxide magnets. In both cases, we find that chiral phonons can
carry giant effective magnetic moments of the order of a Bohr magneton, orders
of magnitude larger than previous predictions.
