  We prove the existence of extensive many-body Hamiltonians with few-body
interactions and a many-body mobility edge: all eigenstates below a nonzero
energy density are localized in an exponentially small fraction of
"energetically allowed configurations" within Hilbert space. Our construction
is based on quantum perturbations to a classical low-density parity check code.
In principle, it is possible to detect this eigenstate localization by
measuring few-body correlation functions in efficiently preparable mixed
states.
