Classification of topological orders

Quantum many-body states such as quantum Hall states, spin liquids, and topological insulators differ from superconductors, superfluids, and anti-ferromagnets in that they do not exhibit spontaneously broken symmetries. The latter is a major organising principle of quantum condensed matter: the broken symmetry can be used to distinguish different states and leads to new low-energy collective excitations (Goldstone bosons).

So, how does one characterise and categorise different states without broken symmetries?

Topological order has been proposed by Xiao-Gang Wen to be the relevant organising principle.
An earlier post considered the role of edge states in such a classification.

How does topology enter?
1. Consider a fractional quantum Hall system on different surfaces with different genus (sphere, torus, connected donuts, ...). Then the ground state is degenerate (in the thermodynamic limit) and the degeneracy depends on the genus of the surface.
In contrast, if one considers a two dimensional non-interacting gas of fermions, there is a unique ground state on both a sphere and a donut (plane with periodic boundary conditions).
2. For a system with an energy gap to the lowest excited state one can have edge states (low energy excitations that are spatially confined to the edge of the sample) and these are described by a topological field theory. [I am hazy on what this means; something like that the coupling constant in the action can have only integer values; these depend only on the topology of the space time.]

At the cake [weekly UQ condensed matter] meeting we are struggling with the paper
Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order

Here are a few of the things I have learnt.

I think this is only about gapped states, i.e, where there is a non-zero energy gap to the lowest excited state.

Broadly the topologically ordered states are divided into two classes, depending on whether they have short- or long-range quantum entanglement. The former means that one can perform a set of spatially localised unitary transformations that map the state into a product (i.e. non-entangled) state.

Class I. Long-range entanglement
Topological order is "stable" (i.e. adiabatically connected) to any perturbation of the Hamiltonian.
Examples: fractional quantum Hall states, chiral spin liquids, Kitaev's toric code, topological Mott insulators. Topological superconductors (e.g., a p+ip state) are in this class but also spontaneously break symmetry too.

Class II. Short-range entanglement
Symmetry-protected topological order. This means only "stable" (i.e. adiabatically connected) to perturbations of the Hamiltonian that preserve a specific symmetry.
Examples: Haldane and AKLT phases of Heisenberg spin-1 antiferromagnetic chains, topological insulators.

The paper goes on to consider how for tensor product states one can define renormalisation group flows that will lead to a fixed point which will reveal a "simpler" wave function that can be classified in terms of the several tensors with many indices [provided the relevant symmetry groups are finite dimensional].

I welcome corrections and clarifications.