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A Many-Body Index for Quantum Charge Transport

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Abstract

We propose an index for pairs of a unitary map and a clustering state on many-body quantum systems. We require the map to conserve an integer-valued charge and to leave the state, e.g. a gapped ground state, invariant. This index is integer-valued and stable under perturbations. In general, the index measures the charge transport across a fiducial line. We show that it reduces to (i) an index of projections in the case of non-interacting fermions, (ii) the charge density for translational invariant systems, and (iii) the quantum Hall conductance in the two-dimensional setting without any additional symmetry. Example (ii) recovers the Lieb–Schultz–Mattis theorem, and (iii) provides a new and short proof of quantization of Hall conductance in interacting many-body systems.

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Notes

  1. The symbols \(\Gamma \) and \(\,\mathrm {d}\Gamma \) (not to be confused with the spatial region \(\Gamma \)) are functors mapping one-particle operators to many-body operators, see e.g. [43].

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Acknowledgements

The authors would like to thank Y. Ogata for inspiring discussions. This research was supported in part by funding from the Simons Foundation and the Centre de Recherches Mathématiques, through the Simons-CRM scholar-in-residence program. The work of S.B. was supported by NSERC of Canada. W.D.R. acknowledges the support of the Flemish Research Fund FWO under Grant G076216N.

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Authors and Affiliations

  1. Department of Mathematics, The University of British Columbia, Vancouver, BC, V6T 1Z2, Canada

    Sven Bachmann

  2. Instituut Theoretische Fysica, KU Leuven, 3001, Leuven, Belgium

    Alex Bols & Wojciech De Roeck

  3. Department of Mathematics, Virginia Tech, Blacksburg, VA, 24061-0123, USA

    Martin Fraas

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  1. Sven Bachmann
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Correspondence to Sven Bachmann.

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Communicated by R. Seiringer

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Bachmann, S., Bols, A., De Roeck, W. et al. A Many-Body Index for Quantum Charge Transport. Commun. Math. Phys. 375, 1249–1272 (2020). https://doi.org/10.1007/s00220-019-03537-x

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