Noncommutative geometry and condensed matter physics

Seminar im Sommersemester 2018

Veranstalter:
Jean Bellissard, Raimar Wulkenhaar

Inhalt:
noncommutative geometry, quantum Hall effect

Techniques from noncommutative differential geometry play a major role in mathematical physics. A prime example is the quantum Hall effect, experimentally discovered in 1980, where the Hall conductance in two-dimensional materials subjected to low temperatures and strong magnetic fields is an integer (later also fractional) multiple of e^2/h. A mathematical theory for this phenomenon which takes into account (a) the independence of material and geometry, (b) insensitivity to rational or irrational magnetic flux per unit cell and (c) stability against disorder was developed by J. Bellissard (1986). The necessary tools include the non-commutative Brillouin zone, its relation to irrational rotation algebras, the K-theory of these C^*-algebras, quantum calculus with Dixmier trace, the noncommutative analogue of the Chern number, through Connes' cyclic cohomology, and the index formula. The seminar talks cover the main steps of this programme and give an outlook to the open problem of the fractional quantum Hall effect.

Literatur:
J. Bellissard, A. van Est & H. Schulz-Baldes, "The noncommutative geometry of the quantum Hall effect", J. Math. Phys. 35 (1994) 5373

Termine:
dienstags 12h30-14h00, N3
Beginn: 17.4.2018

Seminarvorträge

17.04. A renormalization group approach to the universality of Wigner's semi-circle law for random matrices with dependent entries Thomas Krajewski (CPT Marseille)

24.04. Introductory Lecture: Why do we need NCG to describe properties of electrons in a condensate Jean Bellissard

08.05. Basic tool to compare theory with experiment: the Integrated Density of states, the Density of State, the spectral measures, the current-curent correlation. Thermal equilibrium. How to introduce the effect of a magnetic field. It will be explained using the NC-formalism. (pdf) Raimar Wulkenhaar

15.05. --- (NCGOA) ---

29.05. The Gap labeling Theorem and K-theory André Schemaitat

19.06. Transport theory: role of phonons, weak coupling approximation, representation by a random noise. The relaxation time approximation (RTA). Derivation of the Kubo formula. Effectiveness of this formula. The Prodan algorithm.

26.06. The Integer Quantum Hall Effect (IQHE), I. Physics, experimental results. The first Laughlin argument. Problems with localized states. Implementation using NCG. The Kubo-Chern formula in the zero temperature limit and quantization of the transverse conductivity. Defining the localization length.

03.07. The Integer Quantum Hall Effect (IQHE), II. Main result: plateaux of conductivity, quantization, the index formula, the relative index, charge transport. Tools used to prove the results: the four traces way, the Connes formulas in cyclic cohomology, the Dixmier trace and localization length, existence of the Index. Stability under perturbations: electric field, finite volume, dissipation. Non validity of the RTA. Mott hopping transport.

10.07. Topological Insulators. Physics and materials. Experimental results. Edge states and topological protection. The problem of disorder. Various discrete symmetries. The classification into 10 categories, Kitaev's proposal and real K-theory. Connection with Clifford algebra. Clifford modules. Real K-theories, KO, KR, 8-fold Bott periodicity. Stability results, Fredholm indices.

17.07. Interactions: second quantization. Algebraic construction. Introducing the disorder. The Spehner construction of a Hilbert C*-bimodule. Open problems.

<--- Raimar Wulkenhaar
<--- Mathematisches Institut
<--- Fachbereich Mathematik und Informatik
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17.04.	A renormalization group approach to the universality of Wigner's semi-circle law for random matrices with dependent entries	Thomas Krajewski (CPT Marseille)
24.04.	Introductory Lecture: Why do we need NCG to describe properties of electrons in a condensate	Jean Bellissard
08.05.	Basic tool to compare theory with experiment: the Integrated Density of states, the Density of State, the spectral measures, the current-curent correlation. Thermal equilibrium. How to introduce the effect of a magnetic field. It will be explained using the NC-formalism. (pdf)	Raimar Wulkenhaar
15.05.	--- (NCGOA) ---
29.05.	The Gap labeling Theorem and K-theory	André Schemaitat
19.06.	Transport theory: role of phonons, weak coupling approximation, representation by a random noise. The relaxation time approximation (RTA). Derivation of the Kubo formula. Effectiveness of this formula. The Prodan algorithm.
26.06.	The Integer Quantum Hall Effect (IQHE), I. Physics, experimental results. The first Laughlin argument. Problems with localized states. Implementation using NCG. The Kubo-Chern formula in the zero temperature limit and quantization of the transverse conductivity. Defining the localization length.
03.07.	The Integer Quantum Hall Effect (IQHE), II. Main result: plateaux of conductivity, quantization, the index formula, the relative index, charge transport. Tools used to prove the results: the four traces way, the Connes formulas in cyclic cohomology, the Dixmier trace and localization length, existence of the Index. Stability under perturbations: electric field, finite volume, dissipation. Non validity of the RTA. Mott hopping transport.
10.07.	Topological Insulators. Physics and materials. Experimental results. Edge states and topological protection. The problem of disorder. Various discrete symmetries. The classification into 10 categories, Kitaev's proposal and real K-theory. Connection with Clifford algebra. Clifford modules. Real K-theories, KO, KR, 8-fold Bott periodicity. Stability results, Fredholm indices.
17.07.	Interactions: second quantization. Algebraic construction. Introducing the disorder. The Spehner construction of a Hilbert C*-bimodule. Open problems.