  We present the first numerically exact study of self-trapped, a.k.a. soliton,
states of electrons that form in materials with strong quadratic coupling to
the phonon coordinates. Previous studies failed to observe predictions based on
the variational approach in continuum space because soliton states form only
when system parameters are taken to the extreme limit. At the variational
level, we establish that finite-radius solitons emerge through the weak
first-order transition as the coupling strength is increased, and subsequently
collapse to the single-site state through strong first-order transition. Both
transitions transform into smooth crossovers between the light and heavy
polaron states in the full quantum treatment. The most surprising effect not
observed in any other polaron model is non-monotonic dependence of the soliton
effective mass and the residue at strong coupling.
