Approaching Irrational Quantization

Abstract

The integer and fractional quantum Hall effects reveal that quantized physical responses can take integer and fractional values, respectively. This naturally raises the question of whether quantization can extend beyond rational numbers to an irrational value. Here we show that such a quantized response can emerge from the limiting behavior of a finite, experimentally realizable geometric network. We develop a method based on infinite geometric structures, in which a sequence of finite networks rapidly converges toward an infinite limit. Following this approach, we identify a planar network whose normalized resistance converges to the irrational value √3, and realize a finite approximation with high-precision geometric control. The measured normalized resistance is 1.735 ± 0.001 h/e² for a fully occupied Landau level in the designed pattern. Our experimental results extend the concept of quantization beyond rational values and demonstrate that an irrational quantized response can be approached through a finite physical system.

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BibTeX RIS

BibTeX

@misc{wu2026approaching,
  title = {{Approaching Irrational Quantization}},
  author = {Zhihong Wu and Zhimou Chen and Siyuan Zhang and Jinyang Chen and L. N. Pfeiffer and K. W. West and Yang Liu and Xi Lin},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.15753},
  eprint = {2610.15753},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.15753},
}

RIS

TY  - UNPB
TI  - Approaching Irrational Quantization
AU  - Zhihong Wu
AU  - Zhimou Chen
AU  - Siyuan Zhang
AU  - Jinyang Chen
AU  - L. N. Pfeiffer
AU  - K. W. West
AU  - Yang Liu
AU  - Xi Lin
PY  - 2026
DA  - 2026/10/11
PB  - arXiv
DO  - 10.48550/arxiv.2610.15753
UR  - https://arxiv.org/abs/2610.15753
AB  - The integer and fractional quantum Hall effects reveal that quantized physical responses can take integer and fractional values, respectively. This naturally raises the question of whether quantization can extend beyond rational numbers to an irrational value. Here we show that such a quantized response can emerge from the limiting behavior of a finite, experimentally realizable geometric network. We develop a method based on infinite geometric structures, in which a sequence of finite networks rapidly converges toward an infinite limit. Following this approach, we identify a planar network whose normalized resistance converges to the irrational value √3, and realize a finite approximation with high-precision geometric control. The measured normalized resistance is 1.735 ± 0.001 h/e^2 for a fully occupied Landau level in the designed pattern. Our experimental results extend the concept of quantization beyond rational values and demonstrate that an irrational quantized response can be approached through a finite physical system.
ER  -

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