  We propose and theoretically investigate a hybrid system composed of a
crystal of trapped ions coupled to a cloud of ultracold fermions. The ions form
a periodic lattice and induce a band structure in the atoms. This system
combines the advantages of scalability and tunability of ultracold atomic
systems with the high fidelity operations and detection offered by trapped ion
systems. It also features close analogies to natural solid-state systems, as
the atomic degrees of freedom couple to phonons of the ion lattice, thereby
emulating a solid-state system. Starting from the microscopic many-body
Hamiltonian, we derive the low energy Hamiltonian including the atomic band
structure and give an expression for the atom-phonon coupling. We discuss
possible experimental implementations such as a Peierls-like transition into a
period-doubled dimerized state.
