What's the meaning of "topological" in "symmetry protected topological phase"

Question

I am trying to understand the symmetry protected topological phase. Most papers only explain the symmetry but none of them explain the meaning of "topological". My question is : What's the meaning of "topological" in "symmetry protected topological phase"?

This post imported from StackExchange Physics at 2014-04-05 04:39 (UCT), posted by SE-user fangniuwawa

Comments

Perhaps this can (partially) answer your question: physics.stackexchange.com/questions/57179/…. I can provide additional details if necessary.

This post imported from StackExchange Physics at 2014-04-05 04:39 (UCT), posted by SE-user NanoPhys

Answer 1

The follow article directly address your question (which is a very good question):

http://physicsoverflow.org/13018/reconciling-topological-insulators-and-topological-order

The article explains the different meaning of "topological" in "topological insulators" and "topological order".

You observed that most papers only explain the symmetry but none of them explain the meaning of "topological". That is a very good observation. Most papers do not talk about "topological" because the interacting "symmetry protected topological phase" is not topological (ie have trivial topological order and short-range entanglement). We call such short-range entangled phases "topological" because some well known examples of them, such as the Haldane phase and topoloical insulator, were refered as "topological" phases. 

This post imported from StackExchange Physics at 2014-04-05 04:39 (UCT), posted by SE-user Xiao-Gang Wen