Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 303 / 1 / b

Let $z=2d$ be the coordination number of the hypercubic lattice. The mean interaction energy per site is $(Jz/2)m_e{m}_o=Jd,m_e{m}_o$. DefineThe two sublattices each contain half the sites, so the mean-field free energy per site isThe entropy terms are the binary-spin mixing entropies and the positive $J$ favors opposite sublattice magnetizations.

Since $I^{\prime }(m)=\mathrm{artanh}m$, stationarity givesThereforewhere

For $h=g=0$, the expansion $I(m)=-\log 2+m^2/2+O(m^4)$ givesWith $m_{\pm }=(m_e\pm m_o)/2$, one has $m_e{m}_o=m_{+}^2-m_{-}^2$ and $m_e^2+m_o^2=2(m_{+}^2+m_{-}^2)$. HenceThe staggered magnetization $m_{-}$ becomes unstable atproducing the antiferromagnetic transition. The coefficient of the uniform magnetization $m_{+}$ remains positive for every $T>0$, so there is no $m_{+}$ transition.