We investigate the topological consequences of coupling chiral fermions to local, dispersionless phonons. This interaction induces a splitting of the phonon spectrum into three bands: a flat band and two bands with linear dispersion, all of which are degenerate at a nodal point located at zero wavevector. The flat band exhibits vanishing Berry curvature, while the linearly dispersing bands carry nontrivial topological features. Their Berry curvature fields assume a hedgehog-like structure in momentum space, analogous to monopole configurations, and reflect the chirality of the underlying fermionic system. Moreover, the effective phonon response reveals a phonon parity anomaly, observable as a discontinuity in the phonon current. This anomaly originates from the singularities of the fermion Green's function and signals the transfer of topological information from fermions to phonons. Our results demonstrate that phonon currents provide a direct probe of electronic chirality and topological structures.
K. Ziegler (Mon,) studied this question.