Work

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

Year: 2026

Type: preprint

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. We concentrate on systems with a spectral gap, with the chemical potential inside the gap and an invertible Dirac operator $\hat Q$ throughout the Matsubara-frequency domain. The mobility-gap case is not considered here and will be addressed in forthcoming publications.

Source: arXiv (Cornell University)

DOI: 10.48550/arxiv.2606.08868

Authors: Kovyrshin, G., Mekrami, A., Miller, J., Zubkov, M. A., Zuevsky, A.

Institutions: Ariel University, Sidi Mohamed Ben Abdellah University, Sami Shamoon College of Engineering, Czech Academy of Sciences, Institute of Mathematics

Topic: Topological Materials and Phenomena

Subfield: Atomic and Molecular Physics, and Optics

Field: Physics and Astronomy

Domain: Physical Sciences

Open Access status: green

OpenAlex ID: W7164019552

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