  Polarons with different types of electron-phonon coupling have fundamentally
different properties. When the dominant interaction is between the electron
density and lattice displacement, the momentum of the ground state does not
change and the polaron gets exponentially heavy at strong coupling. In
contrast, one-dimensional Peierls/Su-Schrieffer-Heeger (PSSH) polarons with
interaction originating from displacement-modulated hopping feature a shift of
the ground-state momentum to finite values and moderate values of effective
mass as coupling is increased REF[Phys. Rev. Lett. 105, 266605 (2010)]. Based
on Diagrammatic Monte Carlo method, we investigate whether unusual properties
of PSSH polarons depend on the type of the displacement-modulated hopping and
to what degree they survive in higher dimension. We study two different PSSH
models: with bosonic degrees of freedom residing on sites (model A) and bonds
(model B) of the two-dimensional square lattice. For model A, we find that in
both adiabatic and intermediate regimes, the momentum of the ground state
experiences a continuous transition from zero to a finite value as a function
of coupling strength. The transition is driven by quadratic instability of the
dispersion function, implying that effective mass diverges at the critical
point, and then decreases in an anisotropic fashion with increasing coupling.
Unexpectedly, for model B, the momentum of the ground state always stays at
zero and the effective mass increases monotonously with coupling. The increase
is far from exponential and tends to level-off at strong interaction, resulting
in relatively light polarons. Having light polarons in the strong coupling
regime is crucial for the bi-polaron mechanism of high-temperature
superconductivity REF[Phys. Rev. Lett. 121, 247001 (2018)].
