  We study the interplay between the electron-electron (e-e) and the
electron-phonon (e-ph) interactions in the two-orbital Hubbard-Holstein model
at half filling using the dynamical mean field theory. We find that the e-ph
interaction, even at weak couplings, strongly modifies the phase diagram of
this model and introduces an orbital-selective Peierls insulating phase (OSPI)
that is analogous to the widely studied orbital-selective Mott phase (OSMP). At
small e-e and e-ph coupling, we find a competition between the OSMP and the
OSPI, while at large couplings, a competition occurs between Mott and
charge-density-wave (CDW) insulating phases. We further demonstrate that the
Hund's coupling influences the OSPI transition by lowering the energy
associated with the CDW. Our results explicitly show that one must be cautious
when neglecting the e-ph interaction in multiorbital systems, where multiple
electronic interactions create states that are readily influenced by perturbing
interactions.
