Topological invariant responsible for the integer QHE and non-commutative geometry
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. We concentrate on systems with a spectral gap, with the chemical potential inside the gap and an invertible Dirac operator Q̂ throughout the Matsubara-frequency domain. The mobility-gap case is not considered here and will be addressed in forthcoming publications.
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BibTeX
@misc{kovyrshin2026topological,
title = {{Topological invariant responsible for the integer QHE and non-commutative geometry}},
author = {G. Kovyrshin and A. Mekrami and J. Miller and M. A. Zubkov and A. Zuevsky},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.08868},
eprint = {2606.08868},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.08868},
}
RIS
TY - UNPB
TI - Topological invariant responsible for the integer QHE and non-commutative geometry
AU - G. Kovyrshin
AU - A. Mekrami
AU - J. Miller
AU - M. A. Zubkov
AU - A. Zuevsky
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2606.08868
UR - https://arxiv.org/abs/2606.08868
AB - 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 Q̂ throughout the Matsubara-frequency domain. The mobility-gap case is not considered here and will be addressed in forthcoming publications.
ER -