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On Cluster Properties of Classical Ferromagnets in an External Magnetic Field

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Abstract

Correlation functions of ferromagnetic spin systems satisfying a Lee-Yang property are studied. It is shown that, for classical systems in a non-vanishing uniform external magnetic field h, the connected correlation functions decay exponentially in the distances between the spins, i.e., the inverse correlation length (“mass gap”), m(h), is strictly positive. Our proof is very short and transparent and is valid for complex values of the external magnetic field h, provided that \(\mathrm {Re}\, h \not = 0\). It implies a mean-field lower bound on m(h), as \(h \searrow 0\), first established by Lebowitz and Penrose for the Ising model. Our arguments also apply to some quantum spin systems.

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Acknowledgments

We wish to thank both referees for their comments on an earlier version of the manuscript.

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Authors and Affiliations

  1. Institut für Theoretische Physik, ETH Zürich, HIT K 42.3, Wolfgang-Pauli-Strasse 27, 8093, Zürich, Switzerland

    Jürg Fröhlich

  2. Department of Mathematics, University of California, Los Angeles, 520, Portola Plaza, MS 6172, Los Angeles, CA, 90095, USA

    Pierre-François Rodríguez

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  1. Jürg Fröhlich
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  2. Pierre-François Rodríguez
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Correspondence to Pierre-François Rodríguez.

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Dedicated to David Ruelle and Yasha Sinai, on the occasion of their 80th birthday.

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Fröhlich, J., Rodríguez, PF. On Cluster Properties of Classical Ferromagnets in an External Magnetic Field. J Stat Phys 166, 828–840 (2017). https://doi.org/10.1007/s10955-016-1556-2

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  • DOI: https://doi.org/10.1007/s10955-016-1556-2

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