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S_j on $Z^2$ with $S = 1/2$. Prove long-range order in the ground state: the staggered magnetization $m_s = \\lim \\sqrt{\\langle(1/N \\sum_x (-1)^x S_x)^2\\rangle}$ is positive (numerically $m_s$ approximately 0.307). Answer: a proof.","problem_ref":null,"references":"","settled_by":"A proof of positive staggered magnetization, for example by sharpening reflection-positivity bounds.","status_note":"Proven by reflection positivity for $S \\ge 1$ in 2D (Neves and Perez, 1986; Kennedy, Lieb and Shastry, 1988) and for $S = 1/2$ in 3D (Kennedy, Lieb and Shastry, 1988).","title":"Neel order in the spin-$1/2$ square-lattice Heisenberg 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