Third-order phase transition in Landau theory $F=\frac{a}{2}m^2+\frac{u}{4}m^4+\frac{v}{6}m^6-hm$, where $F$ is the free energy, $m$ is the order parameter, $h$ is the external field, $a=a_0(T-T_c)$, and $a_0>0,u>0$ and $v>0$.We know this free energy expansion describes a second-order phase transition. How to write down a free energy such that the transition is a third-order phase transition?
 A: Although third-order transitions are rare to see, but it is not hard to write down Landau theory for third-order transitions. Let $m$ be the order parameter, the free energy simply reads
$$F=a m^4+ b m^6+\cdots$$
with $b>0$, and $a$ is the driving parameter, which triggers a third-order transition at $a_c=0$. To verify, just calculate the saddle point from $\partial_mF=0$ so that
$$m=\left\{\begin{array}{ll}\sqrt{-2a/3b} & a<0,\\
0 & a>0.\end{array}\right.,$$
and hence
$$F=\left\{\begin{array}{ll} 4a^3/9b^2 & a<0,\\
0 & a>0.\end{array}\right.,$$
which obviously becomes singular in the third-order derivative $\partial_a^3 F$ at $a=0$. Following this line of thought, one can play with Landau theory, and write down
$$F=a m^{2(n-1)}+b m^{2n}+\cdots$$
for $n$th-order ($n\geq2$) phase transitions. But I don't think such theoretical construction really meaningful, because the vanishing lower order terms require fine-tuning of the model, and can hardly be realized physically.
However beyond Landau paradigm, third-order transitions can happen in topological quantum phase transitions. One known example is transition between 2D Chern-insulators, described by the following Hamiltonian
$$ H =\sum_{k} c_k^\dagger (\sin k_x\sigma_x+\sin k_y\sigma_y+(\cos k_x+\cos k_y-2+m)\sigma_z)c_k,$$
where $c_k$ is the electron operator of the momentum $k$. The topological mass $m$ is the driving parameter: $m>0$ ($m<0$) corresponds to the topological (trivial) phase, while $m=0$ is critical. It is easy to show that the free energy (at zero temperature) of the fermion system reads $F\simeq\int\mathrm{d}^2k\sqrt{k^2+m^2}\sim -|m|^3$, so $\partial_m^3 F$ is singular at $m=0$, and hence a third-order transition. During this transition, the gap of the Dirac cone (at $k=0$) closes and reopens, leading to $\pm1$ change in the Chern number of the occupied band, which reflects the underlying change of the band topology. However there is neither symmetry broken nor local order parameters involved in this transition, so it is an example of third-order transition that can not be described by the Landau theory. 
