  We calculate the spectral properties of the one-dimensional Holstein and
breathing polarons using the self-consistent Born approximation. The Holstein
model electron-phonon coupling is momentum independent while the breathing
coupling increases monotonically with the phonon momentum. We find that for a
linear or tight binding electron dispersion: i) for the same value of the
dimensionless coupling the quasiparticle renormalization at small momentum in
the breathing polaron is much smaller, ii) the quasiparticle renormalization at
small momentum in the breathing polaron increases with phonon frequency unlike
in the Holstein model where it decreases, iii) in the Holstein model the
quasiparticle dispersion displays a kink and a small gap at an excitation
energy equal to the phonon frequency w0 while in the breathing model it
displays two gaps, one at excitation energy w0 and another one at 2w0. These
differences have two reasons: first, the momentum of the relevant scattered
phonons increases with increasing polaron momentum and second, the breathing
bare coupling is an increasing function of the phonon momentum. These result in
an effective electron-phonon coupling for the breathing model which is an
increasing function of the total polaron momentum, such that the small momentum
polaron is in the weak coupling regime while the large momentum one is in the
strong coupling regime. However the first reason does not hold if the free
electron dispersion has low energy states separated by large momentum, as in a
higher dimensional system for example, in which situation the difference
between the two models becomes less significant.
