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A different perspective on the history of the proof of Hall conductance quantization

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References

  1. 1.

    Michalakis, S. Why is the Hall conductance quantized? Nat. Rev. Phys. 2, 392–393 (2020).

    Article  Google Scholar 

  2. 2.

    Hastings, M. B. & Michalakis, S. Quantization of hall conductance for interacting electrons on a torus. Commun. Math. Phys. 334, 433–471 (2015).

    ADS  MathSciNet  Article  Google Scholar 

  3. 3.

    Avron, J. E. & Seiler, R. Quantization of the Hall conductance for general, multiparticle Schrodinger hamiltonians. Phys. Rev. Lett. 54, 259 (1985).

    ADS  MathSciNet  Article  Google Scholar 

  4. 4.

    Thouless, D. J., Kohmoto, M., Nightingale, M. P. & den Nijs, M. Quantized Hall conductance in a two-dimensional periodic potential. Phys. Rev. Lett. 49, 405 (1982).

    ADS  Article  Google Scholar 

  5. 5.

    Hastings, M. B. Lieb-Schultz-Mattis in higher dimensions. Phys. Rev. B 69, 104431 (2004).

    ADS  Article  Google Scholar 

  6. 6.

    Misguich, G. and Lhuillier, C. Some remarks on the Lieb-Schultz-Mattis theorem and its extension to higher dimensions. Preprint at: https://arxiv.org/abs/cond-mat/0002170 (2000).

  7. 7.

    Oshikawa, M. Commensurability, excitation gap, and topology in quantum many-particle systems on a periodic lattice. Phys. Rev. Lett. 84, 1535 (2000).

    ADS  Article  Google Scholar 

  8. 8.

    Osborne, T. J. Simulating adiabatic evolution of gapped spin systems. Phys. Rev. A 75, 032321 (2007).

    ADS  Article  Google Scholar 

  9. 9.

    Ingham, A. E. A note on Fourier transforms. J. Lond. Math. Soc. 1, 29–32 (1934).

    MathSciNet  Article  Google Scholar 

  10. 10.

    Bravyi, S. & Hastings, M. B. A short proof of stability of topological order under local perturbations. Commun. Math. Phys. 307, 609 (2011).

    ADS  MathSciNet  Article  Google Scholar 

  11. 11.

    Michalakis, S. & Zwolak, J. P. Stability of frustration-free Hamiltonians. Commun. Math. Phys. 322, 277–302 (2013).

    ADS  MathSciNet  Article  Google Scholar 

  12. 12.

    Michalakis, S. Stability of the area law for the entropy of entanglement. Preprint at: https://arxiv.org/abs/1206.6900 (2012).

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Correspondence to Matthew B. Hastings.

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Hastings, M.B. A different perspective on the history of the proof of Hall conductance quantization. Nat Rev Phys 2, 723 (2020). https://doi.org/10.1038/s42254-020-00255-5

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