  The Hubbard-Holstein model describes fermions on a discrete lattice, with
on-site repulsion between fermions and a coupling to phonons that are localized
on sites. Generally, at half-filling, increasing the coupling $g$ to the
phonons drives the system towards a Peierls charge density wave state whereas
increasing the electron-electron interaction $U$ drives the fermions into a
Mott antiferromagnet. At low $g$ and $U$, or when doped, the system is
metallic. In one-dimension, using quantum Monte Carlo simulations, we study the
case where fermions have a long range coupling to phonons, with characteristic
range $\xi$, interpolating between the Holstein and Fr\"ohlich limits. Without
electron-electron interaction, the fermions adopt a Peierls state when the
coupling to the phonons is strong enough. This state is destabilized by a small
coupling range $\xi$, and leads to a collapse of the fermions, and,
consequently, phase separation. Increasing interaction $U$ will drive any of
these three phases (metallic, Peierls, phase separation) into a Mott insulator
phase. The phase separation region is once again present in the $U \ne 0$ case,
even for small values of the coupling range.
