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Entanglement Entropy and Berezin–Toeplitz Operators

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Abstract

We consider Berezin–Toeplitz operators on compact Kähler manifolds whose symbols are characteristic functions. When the support of the characteristic function has a smooth boundary, we prove a two-term Weyl law, the second term being proportional to the Riemannian volume of the boundary. As a consequence, we deduce the area law for the entanglement entropy of integer quantum Hall states. Another application is for the determinantal processes with correlation kernel the Bergman kernels of a positive line bundle: we prove that the number of points in a smooth domain is asymptotically normal.

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Acknowledgements

The authors would like to express their warm gratitude to Benoît Douçot for bringing about this collaboration and for pointing out that the IQH entanglement entropy can be computed in terms of the spectrum of a convenient Toeplitz operator. B.E. also thanks Nicolas Regnault and Semyon Klevtsov for valuable discussions.

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Authors and Affiliations

  1. Institut de Mathématiques de Jussieu-Paris Rive Gauche, IMJ-PRG, Sorbonne Université, CNRS 75005, Paris, France

    Laurent Charles

  2. Laboratoire de Physique Théorique et Hautes Énergies, LPTHE, Sorbonne Université, CNRS 75005, Paris, France

    Benoit Estienne

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  1. Laurent Charles
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  2. Benoit Estienne
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Correspondence to Laurent Charles.

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Communicated by J. Marklof

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Benoit Estienne was supported by Grant Nos. ANR-17-CE30-0013-01 and ANR-16-CE30-0025.

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Charles, L., Estienne, B. Entanglement Entropy and Berezin–Toeplitz Operators. Commun. Math. Phys. 376, 521–554 (2020). https://doi.org/10.1007/s00220-019-03625-y

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  • DOI: https://doi.org/10.1007/s00220-019-03625-y

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