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On the Hohenberg–Mermin–Wagner Theorem and Its Limitations

A Correction to this article was published on 11 March 2019

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Abstract

Just over 50 years ago, Pierre Hohenberg developed a rigorous proof of the non-existence of long-range order in a two-dimensional superfluid or superconductor at finite temperatures. The proof was immediately extended by Mermin and Wagner to the Heisenberg ferromagnet and antiferromagnet, and shortly thereafter, by Mermin to prove the absence of translational long-range order in a two-dimensional crystal, whether in quantum or classical mechanics. In this paper, we present an extension of the Hohenberg–Mermin–Wagner theorem to give a rigorous proof of the impossibility of long-range ferromagnetic order in an itinerant electron system without spin-orbit coupling or magnetic dipole interactions. We also comment on some situations where there are compelling arguments that long-range order is impossible but no rigorous proof has been given, as well as situations, such as a magnet with long range interactions, or orientational order in a two-dimensional crystal, where long-range order can occur that breaks a continuous symmetry.

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  • 11 March 2019

    There was an error in the sentence following Eq. (3).

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Acknowledgements

My understanding of this subject has benefited greatly, over the years, from discussions with Pierre Hohenberg, Paul Martin, and David Nelson. I am grateful to David Nelson also for helpful comments on the manuscript.

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Correspondence to Bertrand I. Halperin.

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Halperin, B.I. On the Hohenberg–Mermin–Wagner Theorem and Its Limitations. J Stat Phys 175, 521–529 (2019). https://doi.org/10.1007/s10955-018-2202-y

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Keywords

  • Two dimensions
  • Long-range order
  • Fluctuations
  • Superfluid