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Cyclic cocycles and quantized pairings in materials science

Emil Prodan
Dec 2022
摘要
The pairings between the cyclic cohomologies and the K-theories of separable$C^\ast$-algebras supply topological invariants that often relate to physicalresponse coefficients of materials. Using three numerical simulations, weexemplify how some of these invariants survive throughout the full Sobolevdomains of the cocycles. These interesting phenomena, which can be explained byindex theorems derived from Alain Connes' quantized calculus, are now wellunderstood in the independent electron picture. Here, we review recentdevelopments addressing the dynamics of correlated many-fermions systems,obtained in collaboration with Bram Mesland. They supply a completecharacterization of an algebra of relevant derivations over the$C^\ast$-algebra of canonical anti-commutation relations indexed by a genericdiscrete Delone lattice. It is argued here that these results already supplythe means to generate interesting and relevant states over this algebra ofderivations and to identify the cyclic cocycles corresponding to the transportcoefficients of the many-fermion systems. The existing index theorems for thepairings of these cocycles, in the restrictive single fermion setting, arereviewed and updated with an emphasis on pushing the analysis on Sobolevdomains. An assessment of possible generalizations to the many-body setting isgiven.
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