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I have thought about topological order for a long time, but I am still confused it.

Roughly speaking in my understanding, the topological state is the eigen-state of some special symmetry such time reversal symmetry and space inversion symmetry, and distinguished from each other by different eigenvalues. Some people say this state has topological order and is protected by the symmetry.

I want to know what occurs during topological transition.

Is it a phase transition? Is there any universal class?

I think it better to understand what is topological order at first.

So, I compare the topological order with Landau's order.

The Landau's order appears from zero to finite value when breaking symmetry in a system. It is a well-known conclusion. I can imagine what occurs when phase transiting. Some part of the system breaks it symmetry at first and has its local Landau's order. However, at the same time, the other part of the system still dose not break symmetry and its local order is zero. Hence, in average, the whole system has its Landau' order parameter by summing all the local orders in the system. This is why the Landau theory is some kind of mean field theory in my opinion.

However, when I want to use analogy to try to understand the topological order, I have been in trouble. The topological properties of a system is global, not local. Hence, I cannot imagine what happens when the system occurs a topological transition. It looks the topological transition suddenly appears and the system changes its eigenvalue at that moment. This process makes me very confused...

I want t know, what is exactly the topological order of a system?

Dose it appears from zero to a finite value or other similar cases when topology changing?

How to determine its value or the level of topological transition?

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