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arXiv · 2606.08868

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

G. Kovyrshin·A. Mekrami·J. Miller·M. A. Zubkov·A. Zuevsky
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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 ${\cal N}_3$ 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 $HC^3$. According to a set of local index theorems the values of ${\cal N}_3$ can be shown to be integer for a limited class of tight - binding models.

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G. Kovyrshin, A. Mekrami, J. Miller, M. A. Zubkov, A. Zuevsky. 2026-06-07. Topological invariant responsible for the integer QHE and non-commutative geometry. https://arxiv.org/abs/2606.08868

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