I was very happy to learn that the work which led to the award of the [2016 Nobel Prize in Physics](https://www.nobelprize.org/nobel_prizes/physics/laureates/2016/) (shared between David J. Thouless, F. Duncan M. Haldane and J. Michael Kosterlitz) uses Topology. In particular, the prize was awarded *"for theoretical discoveries of topological phase transitions and topological phases of matter"*.


Having read both the popular and advanced versions of the scientific background found on the website linked above, I'm left with the impression that the most advance topological concepts used are winding numbers, vector fields and the Poincaré-Hopf index theorem.


I wonder if anyone who reads MO is sufficiently familiar with this work to explain to me why this impression is correct or incorrect. To ask a precise question:



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> Which topological concepts and results are involved in the work which led to the award of the 2016 Nobel Prize in Physics?
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> 


And a follow-up question:



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> Where should a topologist go to read about topological phases of matter and topological phase transitions?
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Topology and the 2016 Nobel Prize in Physics
Roughly speaking: When a system consists of particles interacting strongly, you won't have large movements for any particle unless all of them move together. There is still "quantization", so you'll have jumps in the type of movement (example: the quantum Hall effect). So the topology of the system plays a role. For example, given a collection of electrons with their spins and an external magnetic field, there are alignment issues with how the spins interact with each other and the magnetic field (and that's where your vector fields and winding numbers pop up). But you're not going to really understand the use of the topology without the terminology coming from Solid State physics (alignments of particles in crystals, spin chains, band gaps, etc).


**Granted that, I highly recommend Witten's [*"Three Lectures On Topological Phases Of Matter"*](https://arxiv.org/abs/1510.07698).** He ends up giving a description of our new Nobelist Haldane's model for the quantum Hall effect. He also explains the relation to Chern-Simons theory, which I think is the last remaining "topological concept" that your question is looking for. In a sense, this is analogous to the gauge-theoretic formulation of electrodynamics. **To make "in a sense" rigorous, I recommend Simons' [*"Holonomy, the Quantum Adiabatic Theorem, and Berry's Phase"*](http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.51.2167).** He gives the relation to the work of Thouless on the quantum Hall effect, and the key here is Berry's geometrical phase factor (which for example appears in Yang-Mills theory that describes electrodynamics).


To better bring in the relation to "topological phase", consider the aforementioned collection of particles with spins (a "spin configuration"). You can imagine a region where these spins rotate 360 degrees (a "vortex"). Traversing this path of rotated spins gives you a phase -- the analog is the Aharanov-Bohm effect (the attainment of a "topological phase" to the electron wavefunction as you traverse a non-simply connected region that contains a magnetic field). An example of a "topological phase transition" is that it may be energetically favorible for more or less vortices to appear, as temperature of the system is varied (an analog here is superconductivity where you transition from penetrating magnetic fields to expulsions of the fields, with appearances of vortices inbetween). This is the work of our new Nobelists Kosterlitz and Thouless.


More references for topologists, related to the quantum Hall effect:  

[The Fractional Quantum Hall Effect,
Chern-Simons Theory, and Integral Lattices](http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.51.2167) (Frohlich's 1994 ICM talk)  

[Mathematical Aspects of the Quantum Hall Effect](http://www.mathunion.org/ICM/ICM1994.1/Main/icm1994.1.0075.0105.ocr.pdf) (by Frohlich)


