  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 for the electronic structure.
Consequently, each individual molecule experiences a self-consistent coupling
to the dipoles of all other molecules, which amount to non-negligible values in
the thermodynamic limit (large ensembles). Thus collective vibrational strong
coupling can alter individual molecules strongly for localized "hotspots"
within the ensemble. Moreover, the discovered cavity-induced polarization
pattern possesses a zero net polarization, which resembles a continuous form of
a spin glass (or better polarization glass). Our findings suggest that the
thorough understanding of polaritonic chemistry, requires a self-consistent
treatment of dressed electronic structure, which can give rise to numerous, so
far overlooked, physical mechanisms.
