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

Write $a={\alpha }_2(T)$ and $u={\alpha }_{2n}>0$. At high temperature $a>0$, so $f(m)=a{m}^2+u{m}^{2n}$ has one minimum at $m=0$. At low temperature $a<0$, the origin is a local maximum and two symmetry-related minima appear. The stationary equation isso the equilibrium magnetization isandBecause $a(T)\sim T-T_c$, the two nonzero minima approach zero continuously as $T\uparrow T_c$. The order parameter is continuous while its response becomes singular, so this is a continuous phase transition.

The ordered solution givesso the order-parameter critical exponent isAt $T=T_c$, adding a magnetic term $-hm$ gives $h=2nu{m}^{2n-1}$, henceAbove $T_c$, the zero-field susceptibility is $\chi =(2a{)}^{-1}$. Below $T_c$, the curvature at a minimum is $-4(n-1)a$, so both sides giveAt the ordered minimum, $a=-nu{m}^{2n-2}$ andSince $f_{\mathrm{sing}}\sim |T-T_c{|}^{2-\alpha }$, the heat-capacity critical exponent isThese values obey the Widom scaling relation and Rushbrooke scaling relation at mean-field level.

The Landau-Ginzburg theory promotes the order parameter to a field and penalizes spatial gradients:with stiffness $\kappa >0$. Further even powers and higher-derivative terms may be included when required by accuracy.

Mean field givesThe stated Gaussian fluctuation term isAs $T\to T_c$, mean field is more singular precisely whenorThus the upper critical dimension for the $m^{2n}$ interaction is

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.