Past exam of the condensed matter physics course of the University of Oxford / 2018 / 4 / e / Solution

Choose the positive spontaneous branch for $H_z=0$. Its magnetization starts at $M_s(0)$, decreases with temperature, and reaches zero continuously at $T_c$. It remains zero for $T_c\le T<2T_c$. There is an equally valid negative branch below $T_c$; domain averaging can instead give zero net moment.

A fixed small positive magnetic field intensity selects the positive branch. At zero temperature it still saturates at $M_s(0)$; at intermediate temperature it increases the magnetization relative to the zero-field branch. It rounds the zero-field transition and leaves a positive magnetization above $T_c$, tending in the linear-response region to $C{H}_z/(T-\Theta )$. The magnetization therefore does not vanish at $T_c$ in the nonzero-field curve.

For a concrete labelled sketch, the figure uses the illustrative mean-field theory equationIt shows $h=0$ and a fixed small $h=0.01$. The curve is an illustrative equilibrium equation of state, not a determination of the precise critical exponents of GdN. At $h>0$ no symmetry-breaking transition remains along this temperature sweep.