Abstract
We provide a short proof of the quantization of the Hall conductance for gapped interacting quantum lattice systems on the two-dimensional torus. This is not new and should be seen as an adaptation of the proof of Hastings and Michalakis (Commun Math Phys 334:433–471, 2015), simplified by making the stronger assumption that the Hamiltonian remains gapped when threading the torus with fluxes. We argue why this assumption is very plausible. The conductance is given by Berry’s curvature and our key auxiliary result is that the curvature is asymptotically constant across the torus of fluxes.
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Acknowledgements
We would like to thank Y. Avron for his careful reading of the first version of this manuscript, and for his many comments which helped improve this article. WDR acknowledges the support of the Flemish Research Fund FWO under grant G076216N. AB, MF and WDR have been supported by the InterUniversity Attraction Pole phase VII/18 dynamics, geometry and statistical physics of the Belgian Science Policy.
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Communicated by Vieri Mastropietro.
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Bachmann, S., Bols, A., De Roeck, W. et al. Quantization of Conductance in Gapped Interacting Systems. Ann. Henri Poincaré 19, 695–708 (2018). https://doi.org/10.1007/s00023-018-0651-0
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DOI: https://doi.org/10.1007/s00023-018-0651-0