  Can the properties of the thermodynamic limit of a many-body quantum system
be extrapolated by analysing a sequence of finite-size cases? We present a
model for which such an approach gives completely misleading results: a
translationally invariant, local Hamiltonian on a square lattice with open
boundary conditions and constant spectral gap, which has a classical product
ground state for all system sizes smaller than a particular threshold size, but
a ground state with topological degeneracy for all system sizes larger than
this threshold. Starting from a minimal case with spins of dimension 6 and
threshold lattice size 15 x 15, we show that the latter grows faster than any
computable function with increasing local spin dimension. The resulting effect
may be viewed as a new type of quantum phase transition that is driven by the
size of the system rather than by an external field or coupling strength. We
prove that the construction is thermally robust, opening the possibility that
these effects are accessible to experimental observation.
