  We study a spin-phononic system where diamond defect centers are interfaced
with a quasi-one-dimensional phononic crystal. We show that, a single defect
center, coupled to the phonon modes of a phononic crystal waveguide near the
band gap, can seed its own phononic cavity mode with an exponentially decaying
envelope around the defect center's position. The spin-induced phononic cavity,
with a greatly reduced and tunable mode volume, allows coherent phonon-mediated
interactions between distant spins with a highly tunable range, enabling access
to a variety of long-range interacting spin models. This work opens prospects
for exploring quantum many-body physics and quantum information processing with
defect centers and periodic phononic nanostructures.
