Polaron with quadratic electron-phonon interaction

We present a numerically exact study of polaron with quadratic coupling to the oscillator displacement, or ${X}^{2}$ polaron, using two alternative methodological developments. Our results cover both antiadiabatic and adiabatic regimes entire range electron-phonon ${g}_{2}$, from system's stability threshold at attractive ${g}_{2}=\ensuremath{-}1$ arbitrary strong repulsion ${g}_{2}\ensuremath{\gg}1$. The key properties polarons prove dramatically different their linear counterparts. They (i) are insensitive even large except in limit near instability attraction, (ii) depend only on ratio but electron dispersion dimension space, (iii) feature weak lattice deformations point. direct relevance electrons low densities polar materials, including recent proposals for superconducting states.