2018/12/13 by Bertrand I. Halperin · 3 citations
Mathematics · Physics and Astronomy · #Analytic proof #Ferromagnetism #Impossibility #Order (exchange) #Quantum #Quantum and Classical Electrodynamics #Quantum many-body systems #Spectral Theory in Mathematical Physics #Superconductivity #Superfluidity #cond-mat.stat-mech
paper · pdf · doi:10.1007/s10955-018-2202-y
5 pages, 0 figures. Added reference, and minor corrections. Submitted to the Journal of Statistical Physics, for an issue dedicated to the memory of Pierre Hohenberg
openalex created_date 2018/12/11 · crossref issued 2018/12/13 · crossref published 2018/12/13 · crossref published-online 2018/12/13 · openalex publication_date 2018/12/13 · crossref created 2018/12/13 · arxiv created 2018/12/14 · arxiv updated 2018/12/17 · crossref published-print 2019/05/01 · crossref deposited 2019/12/12 · crossref indexed 2026/07/22 · openalex updated_date 2026/08/05
Just over fifty 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 N. D. Mermin and H. 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.