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The proof works for classical spins and for the antiparallel quantum case, but not for the quantum ferromagnet.","posed_since":"","precise":"H = -sum_<xy> S_x . S_y on $Z^3$ with spin $S = 1/2$ (or any S). Prove that for T below some $T_c > 0$ the infinite-volume Gibbs states have spontaneous magnetization, e.g. $\\lim_{L} L^{-6} \\langle(\\sum_x S_x)^2\\rangle > 0$ in the box of side L. Answer: a proof.","problem_ref":null,"references":"","settled_by":"A proof of long-range order at positive temperature for the spin-$1/2$ quantum Heisenberg ferromagnet on $Z^{3}$.","status_note":"Reflection positivity gives order for the classical ferromagnet (Frohlich, Simon and Spencer, 1976) and the quantum antiferromagnet (Dyson, Lieb and Simon, 1978; Kennedy, Lieb and Shastry, 1988) but fails here; the spin-wave free energy is proven at low T (Correggi, Giuliani and Seiringer, Communications in Mathematical Physics 2015, arXiv 1312.7873); a 2017 preprint claim (Suto, arXiv 1710.04441) lists no journal publication.","title":"Spontaneous magnetization of the 3D quantum Heisenberg 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