The "topological" in "topological order" and the "topological" in "topological insulator" has different meanings.
The 'topological' in topological order means 'robust against ANY local perturbations'.
The "topological" in "topological insulator" means 'robust against some local perturbations that respect certain symmetry'. In fact the properties of symmetry-breaking order are also 'robust against some local perturbations that respect the symmetry'. In this sense, we may also call symmetry-breaking order
"topological" (in the same sense we call topological insulator "topological").
There are two kinds of topology in math. The "topology" in "topological order" is directly related to the first kind of topology in mathematics, as in algebraic topology, homology, cohomology, tensor category, and topological quantum field theory. The "topology" in "topological insulator" is related to the second kind of topology in mathematics, as in mapping class, homotopy, K-theory, etc. The first kind of topology is algebraic, while the second kind of topology is related to the continuous manifold of finite dimensions. We may also say that the first kind of topology is "quantum", while the second kind of topology is "classical".
See Reconciling topological insulators and topological order
This post imported from StackExchange Physics at 2015-02-17 11:25 (UTC), posted by SE-user Xiao-Gang Wen