Wednesday, October 5, 2016

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.


  1. Ross, regarding the Chemistry Nobel: what is your take on this?

    I was a bit surprised, because while these people are undoubtedly good and creative chemists (to have made the "assemblies" that they have synthesized), I wonder about whether this indeed will end up as true "molecular machinery", or whether that whole idea just sells well into glossy journals.
    I mean that it's good work, but I don't see the utilitarian inferences that are being made.

    1. pcs, Thanks for the question.

      I am not a chemist and so am not qualified to comment on the technical achievement. But it does look like a fascinating subject from a pure science point of view.

      I share your concerns. I groan when I read the following from the advanced summary.

      "Compared with the machines that changed our world following the industrial revolution of the nineteenth century, molecular machinery is still in a phase of growth. However, just as the world stood perplexed before the early machines, such as the first electric motors and steam engines, there is the potential for a similar explosive development of molecular machines. In a sense, we are at the dawn of a new industrial revolution of the twenty-first century, and the future will show how molecular machinery can become an integral part of our lives. The advances made have also led to the first steps towards creating truly programmable machines, and it can be envisaged that molecular robotics will be one of the next major scientific areas."

      I find it hard to believe that any of this will happen in our lifetimes.

      I also groaned when I saw that for the physics prize they said topological matter was paving the way for quantum computers.

  2. "The weight of evidence for an extraordinary claim must be proportioned to its strangeness."

    Pierre-Simon Laplace

    "Extraordinary claims require extraordinary evidence"

    Carl Sagan

    1. While I agree with the statements, I think you should explain how this applies here.
      I believe there is no doubt about the results presented by these chemists (assuming you're talking about the chemistry Nobel). The questions I have about it have nothing to do with the quality of the work or the reliability of the conclusions, the questions I have are about the practical implications advertised by the Nobel committee.
      That has nothing to do with evidence and everything to do with (over-)selling otherwise very creative and good chemistry.

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