Topological invariant responsible for the integer QHE and non-commutative geometry

2026/06/30 by G. Kovyrshin, A. Mekrami, J. Miller +2
#cond-mat.mes-hall #hep-th #math-ph #math.MP

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Abstract

We consider a wide class of 2D tight - binding models of solid state physics. These models are, in the most general case, non - homogeneous. The topological invariant \cal N₃ responsible for the quantization of the Hall conductivity, for the specific case of the integer quantum Hall effect in 2D, is expressed through the Wigner transformation of the two-point electron Matsubara Green function. We express this invariant as a pairing of the element of the K^-1 group (generated by the Green function) with the specific element of the cyclic cohomology group HC³. According to a set of local index theorems the values of \cal N₃ can be shown to be integer for a limited class of tight - binding models.

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