I am currently reading the "Quantum Information Meets Quantum Matter -- From Quantum Entanglement to Topological Phase in Many-Body Systems". I am trying to understand the proper definition of topological order. As I understand topological order is a phase of matter where we have a pattern of long range entanglement (LRE), and this can be probed by looking at the topological entanglement entropy $S^{topo}$. For a state that has trivial topological order we should find $S^{topo} =0 $. In the book I mentioned before I found the following definition
-
(Box 7.12 at page 203) Topological orders are stable gapped quantum liquid phases.
In this defintion the term "gapped quantum liquid phase" indicates an equivalence class of gapped Hamiltonians with a well defined thermodynamic limit, where the equivalence relation is given by local unitary transformations (LU). The term "stable" refers to the fact the number of ground states of said Hamiltonians is robust under any local perturbation in the thermodynamic limit. So far I have no problem, but then I also found the following defintion
-
(box 7.23 at page 213) Topologically ordered phases are equivalence classes of LRE gapped quantum liquids under the gSL transformations.
Once again we have the term "gapped quantum liquid" but this time we do not require them to be stable. On the other hand now the equivalence relation is not given by LU but by generalized local stochastic transformations (gSL). gSL transformation are not unitary and if they acts on a "small" region of space they preserve the value of $S^{topo}$ (box 7.25).
If I understand correctly "Topological order" and "Topologically ordered phase" correspond to the same thing, but the two definitions given above do not seem equivalent to me. To make the definitions equivalent to each other I need to require the following:
LU equivalence + robust ground state degeneracy $\sim$ gSL equivalence.
Is this equivalence actually correct? Am I missing some important aspects of the definitions given above? Are the two definitions given above interchangeable when it comes to topological order?
