Mermin-Wagner at the crossover temperature

Mermin-Wagner excludes spontaneous (staggered) magnetization in isotropic ferromagnetic (antiferromagnetic) Heisenberg models at finite temperature in spatial dimensions d ≤ 2. While the proof relies on the (microscopic) Bogoliubov inequality, here we illuminate the theorem from an effective field theory point of view. We estimate the crossover temperature Tc and show that, in weak external fields H, it tends to zero: Tc∝H (d = 1) and Tc ∝ 1/|ln H| (d = 2).