Polaron with quadratic electron-phonon interaction

We present a numerically exact study of a polaron with quadratic coupling to the oscillator displacement, or X2 polaron, using two alternative methodological developments. Our results cover both antiadiabatic and adiabatic regimes and the entire range of electron-phonon coupling g2, from the system's stability threshold at attractive g2 = -1 to arbitrary strong repulsion at g2 >> 1. The key properties of X2 polarons prove dramatically different from their linear counterparts. They (i) are insensitive even to large quadratic coupling except in the antiadiabatic limit near the threshold of instability at attraction, (ii) depend only on the adiabatic ratio but are insensitive to the electron dispersion and dimension of space, and (iii) feature weak lattice deformations even at the instability point. Our results are of direct relevance to the properties of electrons at low densities in polar materials, including recent proposals for their superconducting states.