Past exam of the quantum condensed matter course of the University of Cambridge / 2019 / Section B / 6 / c / Solution

Replace the applied field in the doublet response by the Weiss molecular field $B_E={\mu }_0\lambda M$. At zero applied field,The equation is odd in $M$, so the two ordered orientations have equal magnitude. With $m=|M|/(n{\mu }_B)$, defineThe mean-field self-consistency equation becomesPlot $y=m$ and $y=\tanh (m/t)$. The second has slope $1/t$ at zero and is concave for positive $m$. For $0<t<1$ it starts above the straight line but is below it by $m=1$, giving one positive intersection. For $t\ge 1$, $\tanh (m/t)<m$ for every $m>0$. HenceAt $T=0$ the limiting solution is $m=1$. The onset of spontaneous magnetization is the mean-field Curie temperature.