  Translationally invariant flatband Hamiltonians with interactions lead to a
many-body localization transition. Our models are obtained from single particle
lattices hosting a mix of flat and dispersive bands, and equipped with
fine-tuned two--body interactions. Fine-tuning of the interaction results in an
extensive set of local conserved charges and a fragmentation of the Hilbert
space into irreducible sectors. In each sector, the conserved charges originate
from the flatband and act as an effective disorder inducing a transition
between ergodic and localized phases upon variation of the interaction
strength. Such fine-tuning is possible in arbitrary lattice dimensions and for
any many-body statistics. We present computational evidence for this transition
with spinless fermions.
