At the <first-order phase transition>, the central minimum $m=0$ and two nonzero minima $m=\pm m_0$ are degenerate, with barriers between them. Writing $x=m_0^2>0$, stationarity and equal free energies give
$$
a_2+2a_4x+3a_6x^2=0,
\qquad
a_2+a_4x+a_6x^2=0.
$$
Their difference gives $a_4x+2a_6x^2=0$, so
$$
m_0^2=-\frac{a_4}{2a_6},
\qquad
\boxed{a_2=\frac{a_4^2}{4a_6}}.
$$
The magnitude of the <magnetization> therefore jumps from zero to
$$
\boxed{|m_0|=\sqrt{-\frac{a_4}{2a_6}}}.
$$
The two possible signs are related by the model's spin-reversal symmetry.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 303 / 1 / b / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-303/1/b/solution