Chern-Simons type cross-correlations and geometric Born effective charge of phonons

The interplay between different degrees of freedom in condensed matter systems engenders a rich variety of emergent phenomena. In particular, fermions with non-trivial quantum geometry can generate Chern-Simons (CS)-like terms in effective field theories for different gauge fields. For phonons, such terms can result in chiral phonon splitting. Here, we propose that the local Berry curvature can influence the spectra and dynamics of optical phonons, even in materials with zero Chern number, which we demonstrate with a gapped Dirac model. We identify a previously overlooked CS like cross-correlation between electromagnetic and pseudo-gauge fields in 2+1 dimensions which depends on valley Chern number. It facilitates a direct coupling between phonons and photons by inducing a geometric Born effective charge. This opens up a new route for coherent Raman phonon excitation and quantum geometry probes.