I have been doing some research on Topology in Physics and so I came across this picture

Topological Phase Transition

Source is this link.

Now the way I understood Topology so far is that you can classify specific characteristics of matter depending on their global characteristics. These classes seem to be divided by a topological invariant, so:

If I consider the Spins of Fermions on a surface of any matter the way they are repesented on the given picture, the matter enters a new Topological Phase whenever a new Pair of Vortices appears. But (if what I wrote above is true) this new Topological Phase is defined by another topological invariant than it had beforoe. This invariant would be (from my point of view) the number of Vortices in a system.

To clarify that: The way I understood Topological Phases and Topological Phase Transition, a new number of Vortices means a new Topological Phase. But on the given Picture the "blue" Phase and the "red" phase do have the same Number of Vortices in it (4). The only difference is, that the Tight pair of vortices became two single Vortices. That difference does not make it a new number, does it? So why is this a Topological Phase Transition?

  • $\begingroup$ Think about correlation length of the system. In blue case, there is definitely finite length where it has maximum. On the red case, it is broad $\endgroup$ – kakaz Jan 1 '18 at 16:03
  • $\begingroup$ @kakaz Could u explain that a little bit more? I didn't really understand what u are trying to say $\endgroup$ – Hans-Jürgen Jan 1 '18 at 16:24
  • $\begingroup$ I am suggesting that in tight vertex phase, correlation length of vertex positions are certain, so correlation as function of radius has tight peak. In red phase there's is any such object, just as in ideal gas probably. So this two cases are quite different, so they are different phases. In other words in the blue case vertexes are bound, while in red: free. Probably transfer from blue to red not only needs energy but there may be even gap caused by some bound energy, however it may not be the case. $\endgroup$ – kakaz Jan 1 '18 at 16:31
  • $\begingroup$ To my understanding (the same as yours) there is no global difference between the blue and the red situations, and the topological phase transition claimed in this document is not present, at least because the total winding number of these collections of vortices is the same in both cases. Now it might be that the authors of the document define a topological phase transition using more refined criterion than I, like e.g. correlations length or entanglement length. These two would constitute some really naive criteria to my mind, because they are clearly not global. $\endgroup$ – FraSchelle Jan 2 '18 at 13:50
  • 1
    $\begingroup$ Please see Prof. Wen explanation on mathoverflow mathoverflow.net/questions/251470/… $\endgroup$ – David Bar Moshe Jan 3 '18 at 13:59

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