2016 Nobel Prize in Physics: Topology matters in condensed matter

I was delighted to see this year's Nobel Prize in Physics awarded to Thouless, Haldane, and Kosterlitz 
”for theoretical discoveries of topological phase transitions and topological phases of matter”.

A few years ago I predicted Thouless and Haldane, but was not sure they would ever get it. I am particularly glad they were not bypassed, but rather pushed forward, by topological insulators.

There is a very nice review of the scientific history on the Nobel site.

Here are a few random observations, roughly in order of decreasing importance.

First, it is important to appreciate that there are two distinct scientific discoveries here. They do both involve Thouless and topology, but they really are distinct and so Thouless’ contribution in both is all the more impressive.
The “topological phase transition” concerns the Kosterlitz-Thouless transition which is a classical phase transition (i.e. driven by thermal fluctuations) which is driven by vortices (topological objects,
which can also be viewed as non-linear excitations).
The KT transition and the low temperature phase is remarkably different from other phase transitions and phases of matter. It is a truly continuous transition in that all the derivatives of the free energy are continuous and a Taylor expansion about the critical temperature is not defined.
Yet the superfluid density undergoes a jump at the KT transition temperature.
The low temperature phase has power law correlations with an exponent which is not only irrational but non-universal (i.e. it depends on the coupling constant and temperature).
There are deep connections to quantum phase transitions in one-dimensional systems, e.g. in a spin-1/2 XXZ spin chain, but that is another story.

Topological states of matter are strictly quantum.
Having done the KT transition there is no reason why Thouless would have been led to the formulation of the quantum Hall effect in terms of topological invariants.
That is really an independent discovery. Furthermore, the topology and maths is much more abstract because it is not in real space but involves fibre bundles, Chern numbers, and Berry connections.

All of this phenomena are striking examples of emergence in physics: surprising new phenomena, entities, and concepts.
But, here there is a profound issue about theory preceding experiment.
Almost always emergent phenomena are discovered experimentally and later theory scrambles to explain what is going on.
But, here it seems to be different. KT was predicted and then observed.
The Haldane phase was predicted and then observed in real materials.
When I give my emergent quantum matter talk, I sometimes say: “I can’t think of an example of where a new quantum state of matter was predicted and then observed. Sometimes people give the example of BEC in ultracold atomic gases and of topological insulators but they are essentially non-interacting systems."

On the other hand, it is important to acknowledge that all of this was done with effective Hamiltonians (XY models and Heisenberg spin chains). No one started with a specific material (chemical composition) and then predicted what quantum state it would have without any input from experiment.

The background article helped me better appreciate the unique contributions of Kosterlitz. I was in error not to suggest him before. By himself he worked out the renormalisation group (RG) equations for the transition. Also with Nelson he predicted the universal jump in the superfluid density.
As an aside, it is fascinating that the same RG equations appear in the anisotropic Kondo model and were discovered earlier by Phil Anderson, which was also before Wilson did RG.

The background article also notes how it took a while for Haldane’s 1983 conjecture (that integer spin chains had an energy gap to the lowest excited triplet state) to be accepted, and suggests experiment decided.  It should be pointed out that on the theory side that the numerics was not clear (see e.g., this 1989 review by Ian Affleck) until Steve White developed the DMRG (Density Matrix Renormalisation Group) for one-dimensional quantum many-body systems and laid the matter to rest in 1994 by calculating the energy gap and correlation length to five significant figures!

Later I have some minor sociology comments, but don’t want to spoil all the lovely science in this post.