  We present the first approximation free diagrammatic Monte Carlo study of a
lattice polaron interacting with an acoustic phonon branch through the
deformation potential. Weak and strong coupling regimes are separated by a
self-trapping region where quantum resonance between various possible lattice
deformations is seen in the ground state properties, spectral function, and
optical conductivity. The unique feature of such polaron is the interplay
between long- and short wavelength acoustic vibrations creating a composite
phonon cloud and leading to persistent self-trapping due to the existence of
multiple quasi-stable states. This results in a spectral response whose
structure is much more complex than in any of the previously considered polaron
models.
