  It has gone almost unquestioned that superexchange in the t-J (or Hubbard)
model, and not phonons, is responsible for the unconventional ("d-wave")
pairing symmetry of cuprate superconductors. However a number of advanced
numerical studies have not found superconductivity in the Hubbard model. On the
other hand compelling experimental evidence for a strong electron-phonon
interaction (EPI) has currently arrived. Here I briefly review some
phonon-mediated unconventional pairing mechanisms. In particular the anisotropy
of sound velocity makes the phonon-mediated attraction of electrons non-local
in space providing unconventional Cooper pairs with a nonzero orbital momentum
already in the framework of the conventional BCS theory with weak EPI. In the
opposite limit of strong EPI rotational symmetry breaking appears as a result
of a reduced Coulomb repulsion between unconventional bipolarons. Using the
variational Monte-Carlo method we have found that a relatively weak
finite-range EPI induces a d-wave BCS state also in doped Mott-Hubbard
insulators or strongly-correlated metals. These results tell us that poorly
screened EPI with conventional phonons is responsible for the unconventional
pairing in cuprate superconductors.
