# Does the Kosterlitz–Thouless transition connect phases with different topological quantum numbers?

The Kosterlitz-Thouless transition is often described as a "topological phase transition." I understand why it isn't a conventional Landau-symmetry-breaking phase transition: there is no local symmetry-breaking order parameter on either side of the transition, and all thermodynamic quantities remain continuous (though not analytic) at all derivative orders across the transition. I also understand why it is "topological" in the sense that it involves the binding of vortex topological defects.

But when I think of a "topological phase transition," I think of a phase transition between two phases characterized by different topological quantum numbers, which are topological invariants over the entire phase. For example, the transition between a trivial and topological band insulator corresponds to a change in the Chern number of the band structure (between zero and nonzero). The transition between a topologically trivial and topologically ordered phase corresponds to a change in the topologically protected ground-state degeneracy (between 1 and greater than 1).

Is the KT transition, in which neither side of the transition has either long-range entanglement or symmetry-protected topological order, a topological phase transition in this sense? If so, what are the quantum numbers on both sides of the transition? (I'm not counting trivial answers like "the topological quantum number defined as 0 if vortices are bound and 1 if they are unbound.")

• Xiao-Gang Wen's answer over at Math Overflow may be helpful: mathoverflow.net/questions/251470/… "TK transition is a transition between non-topological phases induced by topological defects (vortexes)" – Rococo Dec 20 '16 at 4:42
• ...Which to me looks like you are not missing anything and there are indeed two rather different sense of "topological phase transition" in use. I suspect that this confusion is at least partly the result of the Nobel committee, who looked at several distinct advances all involving Thouless and all invoking topology in some sense, and tried to summarize it as "theoretical discoveries of topological phase transitions and topological phases of matter" – Rococo Dec 20 '16 at 4:45

We may use disordered symmetry breaking states (which are described by non-linear $\sigma$-models) to realize a large class of topological orders. ... ... 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. Thus, it may be confusing to refer 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.
More precisely, if the fluctuating order parameter in a disordered state has no topological defects, then the corresponding disordered state will usually have a non-trivial topological order. The type of the topological order depends on the topology of the degenerate manifold $K$ of the order parameter (ie the target space of the non-linear $\sigma$-model). For example, if $\pi_1(K)$ is a finite group and $\pi_{n>1}(K)=0$, then the disordered phase may have a topological order described by a gauge theory of gauge group $G=\pi_1(K)$. If $\pi_1(K),\pi_2(K)$ are finite groups and $\pi_{n>2}(K)=0$, then the disordered phase may have a topological order described by a 2-gauge theory of 2-gauge-group $B(\pi_1(K),\pi_2(K))$.