  We present exact diagonalization and momentum average approximation (MA)
results for the single polaron properties of a one-dimensional two-band model
with phonon-modulated hopping. At strong electron-phonon coupling, we find a
novel type of sharp transition, where the polaron ground state momentum jumps
discontinuously from $k=\pi$ to $k=0$. The nature and origin of this transition
is investigated and compared to that of the Su-Schrieffer-Heeger (SSH) model,
where a sharp but smooth transition was previously reported. We argue that such
discontinuous transitions are a consequence of the multi-band nature of the
model, and are unlikely to be observed in one-band models. We also show that MA
describes qualitatively and even quantitatively accurately this polaron and its
transition. Given its computationally efficient generalization to higher
dimensions, MA thus promises to allow for accurate studies of electron-phonon
coupling in multi-band models in higher dimensions.
