2016.10.05 |

This year's Nobel prize in physics was given to Thouless, Haldane and Kosterlitz

for their work on Topological phase transitions and topological phases of matter. As stated in the press release:

https://www.nobelprize.org/nobel_prizes/physics/laureates/2016/advanced.html

The whole thing started with Thouless' understanding of why the Hall conductance must be an integer number times the Klitzing constant. In mathematical terms, he proved that this number is the first Chern class of a line bundle (of which the Laughlin states are sections) over a torus and since the Chern class can only take on integer values (for topological reasons) this gave a clear mathematical proof of this phenomenon. From a modern perspective, this is simply just a manifestation of the fact that integer quantum Hall effect is described by an effective theory which is abelian Chern-Simons theory, where the states are holomorphic sections of line bundles over U(1)-moduli spaces of surfaces (ref. [5], [33], [41] in the link above).

This start was followed by further work by Kosterlitz and Haldane, who investigated the phenomenon and proposed topological phase of matter, described by various topological properties of the quantum wave-functions of the systems. Moreover they proposed and described in certain cases the topological phase transition. Haldane also described a topological phase which is now called a "Chern insulator", * *something which has only recently been experimentally verified by a team of Chinese researchers (ref. [15] in the link above).

Today this theory has seen considerable developments on the mathematical side. On the one hand the classification of the possible theoretical classes of topological phases of matter has been pushed forward and it is related to classical algebraic topology and homotopy theory. On the other hand, the branch of mathematics which is concerned with properties of a given topological phase of matter, is new and goes by the name Quantum Topology - a central focus for QGM. Quantum Topology includes quantisation of moduli space and the associated topological quantum field theories, which arises this way, such as quantum Chern-Simons theory for non-abalian groups. These describe very special properties which materials of such topological classes must have.

This is also discussed in section 6 in the above linked description, where possible future ramifications of topological phases are mentioned. In particular the topological quantum computing proposal of on the one hand Kitaev and on the other Freedman & collaborators are mentioned (ref. [32] and [42]). They propose a topological quantum field theory (TQFT) like quantum Chern-Simons theory for non-abelian groups like SU(2) as a universal quantum computer, if it can be realised in the laboratory. In fact this was proposed by Witten (former Nielsen lecturer at QGM) in the early nineties and constructed mathematically rigorously by Reshetikhin (former Niels Bohr professor at QGM) and Turaev (long term visitor at QGM) shortly there after.

Even before its mathematical construction, it was realised that this quantum Chern-Simons theory is closely related to the fractional quantum Hall effect (FQHE) (ref. [41]). Topological quantum field theories (TQFT) are interesting to study from a mathematical point of view since they have a precise mathematical definition (a short video explaining this can be found at https://www.youtube.com/watch?v=oYU4G5ftB0c). That Quantum Chern-Simons theory and TQFT's are also related to topological phases in the FQHE is a further motivation for our intense study of this theory at QGM, where we have established several asymptotic properties, such as the important asymptotic faithfulness of the representations of the mapping class groups and further proved the equivalence between the combinatorial and quantum geometric construction of these theories (see e.g. http://annals.math.princeton.edu/2006/163-1/p07, http://link.springer.com/article/10.1007%2Fs00222-014-0555-7, http://www.ems-ph.org/journals/show_issue.php?issn=1663-487X&vol=3&iss=3)

Very recently, Vafa has proposed SL(2,C) quantum Chern-Simons theory as a good model for the FQHE https://arxiv.org/pdf/1511.03372.pdf. This theory is much less studied mathematically, but we have recently at QGM (joint work between centre director Jørgen Ellegaard Andersen and Rinat Kashaev, Geneva) given a precise mathematical definition of this theory https://arxiv.org/pdf/1409.1208v1.pdf, and its properties are under intens investigation at the moment https://arxiv.org/abs/1608.06872, https://arxiv.org/abs/1409.1035, https://arxiv.org/abs/1608.01761.

Going forward, researchers at Aarhus University are exploring the suggestions of Centre director Jørgen Ellegaard Andersen: https://www.youtube.com/watch?v=BnilIKN63to to use these same mathematical tools to understand firstly folding of proteins and secondly to search for proteins' quantum topological phases.