Acoustical polaron in three dimensions: The ground-state energy and the self-trapping transition

The interaction of an electron with acoustical phonons by the deformation potential is studied with the Feynman path-integral method for zero temperature. An upper bound to the polaron ground-state energy is obtained. The nature of the transition of the quasifree to the self-trapped electron state is discussed for different approximations to the polaron ground-state energy. We find that, within the Feynman approximation, which is the most reliable one for the ground-state energy, there exists a critical value ($k_0^{\mathrm{*}}$) for the cutoff ($k_0$) in phonon wave-vector space such that for $k_0$<$k_0^{\mathrm{*}}$ ($k_0$>$k_0^{\mathrm{*}}$) the self-trapping transition is continuous (discontinuous) as a function of the electron-phonon coupling strength.