  We study the long-time evolution of a quantum memory coupled to a bosonic
environment on which quantum error correction (QEC) is performed using the
surface code. The memory's evolution encompasses $N$ QEC cycles, each of them
yielding a non-error syndrome. This assumption makes our analysis independent
of the recovery process. We map the expression for the time evolution of the
memory onto the partition function of an equivalent statistical mechanical spin
system. In the superohmic dissipation case the long-time evolution of the
memory has the same behavior as the time evolution for just one QEC cycle. For
this case we find analytical expressions for the critical parameters of the
order-disorder phase transition of an equivalent spin system. These critical
parameters determine the threshold value for the system-environment coupling
below which it is possible to preserve the memory's state.
