Défense et illustration de l'intérêt de la géométrie non commutative dans le cadre de la physique du solide...

...celle de l'effet Hall Quantique entier hier

Dans ce billet, le blogueur retourne à ces premières amours, la physique du solide en l’occurrence, ne serait-ce que pour montrer au lecteur qu'il n'y a pas que le boson de Higgs et la physique des particules dans la vie!

In 1880, Hall undertook the classical experiment which led to the so-called Hall effect. A century later, von Klitzing and his co-workers showed that the Hall conductivity was quantized at very low temperatures as an integer multiple of the universal constant e2/h. Here e is the electron charge whereas h is Planck’s constant. This is the integer quantum Hall effect (IQHE). For this discovery, which led to a new accurate measurement of the fine structure constant and a new definition of the standard of resistance, von Klitzing was awarded the Nobel price in 1985. 

After the works by Laughlin and especially by Kohmoto, den Nijs, Nightingale, and Thouless’ (called TKN, below), it became clear that the quantization of the Hall conductance at low temperature had a geometric origin. The universality of this effect had then an explanation. Moreover, as proposed by Prange, Thouless, and Halperin, the plateaux of the Hall conductance which appear while changing the magnetic field or the charge-carrier density, are due to localization. Neither the original Laughlin paper nor the TKN one however could give a description of both properties in the same model. Developing a mathematical framework able to reconcile topological and localization properties at once was a challenging problem. Attempts were made by Avron et al. who exhibited quantization but were not able to prove that these quantum numbers were insensitive to disorder. In 1986, Kunz went further on and managed to prove this for disorder small enough to avoid filling the gaps between Landau levels. 

But ... [Bellissard] proposed to use noncommutative geometry to extend the TKN, argument to the case of arbitrary magnetic field and disordered crystal. It turned out that the condition under which plateaux occur was precisely the finiteness of the localization length near the Fermi level. This work was rephrased later on by Avron et aLI7 in terms of charge transport and relative index, filling the remaining gap between experimental observations, theoretical intuition and mathematical frame.

J. Bellissard, A. van Elst et H. Schulz- Baldest, The noncommutative geometry of the quantum Hall effect, 10/10/1994

...celle des isolants topologiques aujourd'hui

It is often said that the topological insulators (TI) are the equivalent of the Quantum Integer Hall Effect (IQHE), but without the need of an external magnetic field (or any other external field for that matter)...  the stability of topological phases under strong disorder should be placed among the key issues in an complete theory of TIs....an important observation is the following simple but fundamental principle: If the bulk topological invariant, classifying the different phases of a TI, stays quantized and non-fluctuating as long as the Anderson localization length is finite, then the characteristics discussed above are necessarily present. Indeed, this property will ensure, on one hand, the stability of the topological phases in the presence of strong disorder and, on the other hand, that the only way to cross from one topological phase to another is via a divergence of the localization length. For IQHE, the Hall conductance as proven to posses this property using non-commutative geometry in the 1990’s [9]. This result represents one of the most important applications of the non-commutative geometry in condensed matter physics (cf [1] pg. 365). It was only recently that similar mathematically rigorous results start appearing for other topological phases [2, 3]. They gave a complete characterization of the entire complex classes (the A- and AIII-symmetry classes in any dimension) of the periodic table [5] of topological insulators and superconductors. As such, the methods of non-commutative geometry have been extended from the upper-left corner of this table to the entire rows of the complex classes. 

When discussing these results with his colleagues, the author is often asked how crucial is Alain Connes’ non-commutative geometry for the whole developmen? Of course, after understanding the arguments and seeing the final conclusions, one can reproduce them via  different methods. However, without the guidance from noncommutative geometry, searching for the correct form of the index theorems would have been like searching for a needle in a haystack. To convince the reader of this fact, the paper presents first some key elements of the noncommutative geometry program, which as we shall see lay down the basic principles and the guiding philosophy. Then the paper gradually builds the specific structures needed for the problem at hand. The quantization and the homotopy stability of the topological invariants, together with the general conditions when these happen, will then naturally emerge... 

If the author is allowed to share some thoughts about his experience, then these will be his words for the noncommutative geometry as applied to materials science:

• The framework provides guidance and intuition. When done inside this framework, the search for the correct invariants no longer feel like searching for a needle in a haystack.

•  It provides the big picture so one can always know what he is computing. In the present context, the index theorems we just presented give morphisms from the K-groups of algebra of localized observables into the ℤ. 

• Last but not the least, the framework provides some outstanding tools of calculus. It will be a true asset to the materials science if a wider acceptance is achieved among the physicists. Needles to say, the field of Topological Insulators is the perfect ground for applications.

 Emil Prodan , The Non-Commutative Geometry of the Complex Classes of Topological Insulators, 28/02/2014