  Recently, it has become apparent that the thermal stability of topologically
ordered systems at finite temperature, as discussed in condensed matter
physics, can be studied by addressing the feasibility of self-correcting
quantum memory, as discussed in quantum information science. Here, with this
correspondence in mind, we propose a model of quantum codes that may cover a
large class of physically realizable quantum memory. The model is supported by
a certain class of gapped spin Hamiltonians, called stabilizer Hamiltonians,
with translation symmetries and a small number of ground states that does not
grow with the system size. We show that the model does not work as
self-correcting quantum memory due to a certain topological constraint on
geometric shapes of its logical operators. This quantum coding theoretical
result implies that systems covered or approximated by the model cannot have
thermally stable topological order, meaning that no system can be stable
against both thermal fluctuations and local perturbations simultaneously in two
and three spatial dimensions.
