What is so topological about topological phase transitions?

Question

I am studying the KT-transition, which is called a topological phase transition. The phase transition is driven by vortices in a 2-D superfluid, where it is shown that at a critical temperature $T_c$ free vortices are energetically favoured over vortex-antivortex pairs.

There is no spontaneous symmetry breaking in this phase transition. Instead it is called a topological phase transition. However, in my very naive understanding of topolopy (obtained from looking at donuts and pretzels and stuff), we are interested in the structure, or the singularities of the system. Even in the case of the transition where these dislocations/vortices/singularities become free, the number of singularities doesn't necessarily change. I would say that the topology is conserved, since only the distance between the singularities is changed.

If the topology of the system doesn't change, why do we call it a topological phase transition?

Answer 1

Let us consider a quantum phase transition (at T=0) from an ordered phase to a disordered phase, driven by the quantum fluctuations of the order parameter. We like to ask if the disordered phase has topological order or not.

The importance of the topological defects in phase transitions have been emphasized by Kosterlitz and Thouless, who shared 2016 Nobel prize (with Haldane) ``for theoretical discoveries of topological phase transitions and topological phases of matter''.

In this paper, we show that the phase transitions driven by fluctuations with all possible topological defects produce disordered states that have no topological order, and correspond to non-topological phase transitions. While transitions driven by fluctuations without any topological defects usually produce disordered states that have non-trivial topological orders, and correspond to topological phase transitions. If we refer to phase transitions induced by topological defects as topological phase transitions, and refer to phase transitions between different topological orders as topological phase transitions, then our result can be restated as:

in general topological phase transitions are not topological phase transitions, nor are topological phase transitions topological phase transitions.

Thus, it may be confusing to refer to the transition driven by topological defects as a topological phase transitions, since the appearance of topological defects decrease the chance to produce topological phases of matter.