Title: Unraveling a cavity induced molecular polarization mechanism from collective vibrational strong coupling

Abstract: We demonstrate that collective vibrational strong coupling of molecules in thermal equilibrium can give rise to significant local electronic polarizations in the thermodynamic limit. We do so by first showing that the full non-relativistic Pauli-Fierz problem of an ensemble of strongly-coupled molecules in the dilute-gas limit reduces in the cavity Born-Oppenheimer approximation to a cavity-Hartree equation. Consequently, each molecule experiences a self-consistent coupling to the dipoles of all other molecules. The here derived cavity-Hartree equations allow for a computationally efficient implementation in an ab-initio molecular dynamics setting. For a randomly oriented ensemble of slowly rotating model molecules, we observe a red shift of the cavity resonance due to the polarization field, which is in agreement with experiments. We then demonstrate that the back-action on the local polarization takes a non-negligible value in the thermodynamic limit and hence the collective vibrational strong coupling can polarize individual molecules. The observed local polarization pattern with zero net polarization resembles a continuous form of a spin-glass (or better polarization-glass) phase. The continuous polarization distribution implies the existence of hotspots within the molecular ensemble, where the collective coupling can strongly alter local molecular properties. For atomic ensembles, however, these local polarization mechanism is absent, since room temperature cannot induce any disorder in the dilute limit. Overall, our findings suggest that the thorough understanding of polaritonic chemistry, such as the modification of chemical reactions, requires a self-consistent treatment of the cavity induced polarization and the usually applied restrictions to the displacement field effects are insufficient.