  We propose and study a model of a quantum memory that features
self-correcting properties and a lifetime growing arbitrarily with system size
at non-zero temperature. This is achieved by locally coupling a 2D L x L toric
code to a 3D bath of bosons hopping on a cubic lattice. When the stabilizer
operators of the toric code are coupled to the displacement operator of the
bosons, we solve the model exactly via a polaron transformation and show that
the energy penalty to create anyons grows linearly with L. When the stabilizer
operators of the toric code are coupled to the bosonic density operator, we use
perturbation theory to show that the energy penalty for anyons scales with
ln(L). For a given error model, these energy penalties lead to a lifetime of
the stored quantum information growing respectively exponentially and
polynomially with L. Furthermore, we show how to choose an appropriate coupling
scheme in order to hinder the hopping of anyons (and not only their creation)
with energy barriers that are of the same order as the anyon creation gaps. We
argue that a toric code coupled to a 3D Heisenberg ferromagnet realizes our
model in its low-energy sector. Finally, we discuss the delicate issue of the
stability of topological order in the presence of perturbations. While we do
not derive a rigorous proof of topological order, we present heuristic
arguments suggesting that topological order remains intact when perturbative
operators acting on the toric code spins are coupled to the bosonic
environment.
