Bellissard, Jean \(C^*\) algebras in solid state physics. 2D electrons in a uniform magnetic field. (English) Zbl 0677.46055 Operator algebras and applications. Vol. 2: Mathematical physics and subfactors, Pap. UK-US Jt. Semin., Warwick/UK 1987, Lond. Math. Soc. Lect. Note Ser. 136, 49-76 (1988). Summary: [For the entire collection see Zbl 0668.00015.] Technics recently developed in non commutative geometry through properties of \(C^*\) algebras are presented without proofs here to investigate some properties of 2D electrons in a uniform magnetic field. The Peierls substitution, a commonly used approximation for Bloch electrons, is justified and leads to a rotation algebra. A new differential calculus, analogous to the Itô calculus for stochastic processes, is introduced to investigate the fine structure of the energy spectrum. We announce the proof of the Wilkinson Rammal formula accordng to which the derivative of the energy gap boundaries are discontinuous at each rational values of the magnetic flux. We also announce that the derivative is continuous at irrational values of this flux. At last we review and improve the results previously obtained for the quantum Hall effect and sketch the proof that in the region of localized states the Hall conductance exhibits plateaux at integer values of the universal constant \(e^ 2/h\). Cited in 1 ReviewCited in 22 Documents MSC: 46L60 Applications of selfadjoint operator algebras to physics 46L55 Noncommutative dynamical systems 81V10 Electromagnetic interaction; quantum electrodynamics 46L80 \(K\)-theory and operator algebras (including cyclic theory) 46N99 Miscellaneous applications of functional analysis 81V70 Many-body theory; quantum Hall effect Keywords:non commutative geometry through properties of \(C^*\) algebras; 2D electrons in a uniform magnetic field; Peierls substitution; Bloch electrons; rotation algebra; differential calculus, analogous to the Itô calculus for stochastic processes; Wilkinson Rammal formula; quantum Hall effect; localized states; Hall conductance Citations:Zbl 0668.00015 × Cite Format Result Cite Review PDF