Symmetry breaking and long range order in Heisenberg antiferromagnets

We prove the bound ${\mathit{m}}_{\mathit{s}}$(β)≥ √3 σ(β) between the spontaneous staggered magnetization ${\mathit{m}}_{\mathit{s}}$(β) and the long range order parameter σ(β) in the quantum Heisenberg antiferromagnets. The same bound has been known for the Heisenberg ferromagnets from the works of Griffiths, and of Dyson, Lieb, and Simon, but extensions to the antiferromagnets had been lacking for more than two decades. When combined with the results on nonvanishing σ(β) from the Dyson-Lieb-Simon method, our bound proves the existence of symmetry breaking in the physically natural equilibrium states and ground states obtained by applying an infinitesimal symmetry breaking field.