Landau theory

Remarks

Example: at $h = 0$, set $a = a_0 (T - T_c)$. The equation $2 a m + 4 b m^3 = 0$ and the sign of $f_L$ give $m = 0$ for $T \ge T_c$ and $m = \pm \sqrt{a_0 (T_c - T) / (2b)}$ for $T < T_c$. So $m \sim (T_c - T)^{1/2}$, which is the critical exponent $\beta = 1/2$. In the same way, the susceptibility $\partial m / \partial h$ at $h = 0$ is $1/(2a)$ above $T_c$ and $1/(4\abs{a})$ below, so $\gamma = 1$. At $T = T_c$, $h = 4 b m^3$, so $\delta = 3$, and the free energy $-a^2/(4b)$ below $T_c$ gives a jump of the specific heat, so $\alpha = 0$.

Non-example: a free energy with a cubic term $c\, m^3$, $c \ne 0$, fails clause 2. Such a term is allowed when no symmetry forbids it, and it makes the transition first order, with a jump of $m$.

For a vector order parameter in $\R^n$ with symmetry $O(n)$, $m^2$ and $m^4$ are replaced by $\abs{m}^2$ and $\abs{m}^4$.