  We discuss and review several thermodynamic criteria that have been
introduced to characterize the thermal stability of a self-correcting quantum
memory. We first examine the use of symmetry-breaking fields in analyzing the
properties of self-correcting quantum memories in the thermodynamic limit: we
show that the thermal expectation values of all logical operators vanish for
any stabilizer and any subsystem code in any spatial dimension. On the positive
side, we generalize the results in [R. Alicki et al., arXiv:0811.0033] to
obtain a general upper bound on the relaxation rate of a quantum memory at
nonzero temperature, assuming that the quantum memory interacts via a Markovian
master equation with a thermal bath. This upper bound is applicable to quantum
memories based on either stabilizer or subsystem codes.
