  We study a single polaron in the Su-Schrieffer-Heeger (SSH) model using four
different techniques (three numerical and one analytical). Polarons show a
smooth crossover from weak to strong coupling, as a function of the
electron-phonon coupling strength $\lambda$, in all models where this coupling
depends only on phonon momentum $q$. In the SSH model the coupling also depends
on the electron momentum $k$; we find it has a sharp transition, at a critical
coupling strength $\lambda_c$, between states with zero and nonzero momentum of
the ground state. All other properties of the polaron are also singular at
$\lambda = \lambda_c$, except the average number of phonons in the polaronic
cloud. This result is representative of all polarons with coupling depending on
$k$ and $q$, and will have important experimental consequences (eg., in ARPES
and conductivity experiments).
