Emergent long-range interactions in Bose-Einstein Condensates
  We consider a massive complex scalar field with contact interactions with a
source and show that, upon Bose-Einstein condensation, there is an emergent
long-range interaction between sources. This interaction becomes long-range in
the limit of vanishing self-interaction between Bose-Einstein constituents.
More generally, the range is given by $\ell^{-1}\propto \sqrt{\lambda n/m}$,
with $\lambda$ being the 2-body self-interaction coupling constant, $n$ the
particle number density in the condensate, and $m$ the mass of the condensed
particles. Naively this may sound surprising since in $\lambda\rightarrow 0$
limit gapless excitations of the condensate have dispersion relation
$\omega_k=k^2/2m$, yet for the mediated force we have $F\propto 1/r^2$. The
reason behind this seemingly counterintuitive result lies in the fact that the
force is being mediated by the phonon, which happens to acquire a nontrivial
derivative interaction with the source. We discuss the potential ramifications
of this observation for dark matter models. In particular, we show that this
force can compete with gravity on galactic scales for a wide range of dark
matter mass, provided that the interaction with baryons allows the presence of
an extended dark matter condensate core. The effect could be of particular
interest in ultra-light dark matter models, such as Fuzzy Dark Matter.
