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Entanglement in the Quantum Ising Model

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Abstract

We study the asymptotic scaling of the entanglement of a block of spins for the ground state of the one-dimensional quantum Ising model with transverse field. When the field is sufficiently strong, the entanglement grows at most logarithmically in the number of spins. The proof utilises a transformation to a model of classical probability called the continuum random-cluster model, and is based on a property of the latter model termed ratio weak-mixing. In an intermediate result, we establish an exponentially decaying bound on the operator norm of differences of the reduced density operator. Of special interest is the mathematical rigour of this work, and the fact that the proof applies equally to a large class of disordered interactions.

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Authors and Affiliations

  1. Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WB, UK

    Geoffrey R. Grimmett

  2. Department of Mathematics, Royal Holloway, University of London, Egham, Surrey, TW20 0EX, UK

    Tobias J. Osborne

  3. Scuola Internazionale Superiore di Studi Avanzati, via Beirut 2-4, 34014, Trieste, Italy

    Petra F. Scudo

  4. INFN, Sezione di Trieste, Trieste, Italy

    Petra F. Scudo

Authors
  1. Geoffrey R. Grimmett
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  2. Tobias J. Osborne
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  3. Petra F. Scudo
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Correspondence to Geoffrey R. Grimmett.

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Grimmett, G.R., Osborne, T.J. & Scudo, P.F. Entanglement in the Quantum Ising Model. J Stat Phys 131, 305–339 (2008). https://doi.org/10.1007/s10955-008-9502-6

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  • DOI: https://doi.org/10.1007/s10955-008-9502-6

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