We generalize the notion of dissipationless, topological Hall viscosity tensor to optical phonons in thin film Weyl semimetals. By using the strained Porphyrin thin film Weyl semimetal as a model example, we show how optical phonons can couple to Wey
l electrons as chiral pseudo gauge fields. These chiral vector fields lead to a novel dissipationless two-rank viscosity tensor in the effective dynamics of optical phonons whose origin is the chiral anomaly. We also compute the contribution to this two rank Hall viscosity tensor due to the presence of an external magnetic field, whose origin is the conventional Hall response of Weyl electrons. Finally, the phonon dispersion relations of the system at the long-wavelength limit with and without an electromagnetic field are calculated showing a measurable shift in the Raman response of the system. Our results can be investigated by Raman scattering or infrared spectroscopy by attenuated total reflectance experiments.
