  In [Phys. Rev. A 88, 062313 (2013)] we proposed and studied a model for a
self-correcting quantum memory in which the energetic cost for introducing a
defect in the memory grows without bounds as a function of system size. This
positive behavior is due to attractive long-range interactions mediated by a
bosonic field to which the memory is coupled. The crucial ingredients for the
implementation of such a memory are the physical realization of the bosonic
field as well as local five-body interactions between the stabilizer operators
of the memory and the bosonic field. Here, we show that both of these
ingredients appear in a low-energy effective theory of a Hamiltonian that
involves only two-body interactions between neighboring spins. In particular,
we consider the low-energy, long-wavelength excitations of an ordered
Heisenberg ferromagnet (magnons) as a realization of the bosonic field.
Furthermore, we present perturbative gadgets for generating the required
five-spin operators. Our Hamiltonian involving only local two-body interactions
is thus expected to exhibit self-correcting properties as long as the noise
affecting it is in the regime where the effective low-energy description
remains valid.
