  Nieh-Yan anomaly describes the non-conservation of chiral charges induced by
the coupling between Dirac fermions and torsion fields. Since the torsion field
is beyond general relativity, this effect remains hypothetical and its
relevance to our universe is unclear in the context of high-energy physics. In
this work, we propose that the phonons can induce a torsion field for the
Kramers-Weyl fermions through electron-phonon interaction in a non-magnetic
chiral crystal, thus leading to the occurrence of the Nieh-Yan anomaly in this
condensed matter system. As a consequence, the Nieh-Yan term can strongly
influence the phonon dynamics and lead to the helicity of acoustic phonons,
namely, two transverse phonon modes mix with each other to form a circular
polarization with a non-zero angular momentum at a finite phonon momentum and
the phonon angular momentum reverses its sign for opposite momenta due to time
reversal symmetry. The phonon helicity can be probed through measuring the
total phonon angular momentum driven by a temperature gradient.
