Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 303 / 1

Landau theory treats the order parameter as spatially uniform and expands the free-energy density in powers allowed by its symmetries. The Landau-Ginzburg theory promotes it to a field $\varphi (\mathbf{x})$ and adds gradient terms such as $|\nabla \varphi {|}^2$. It therefore describes spatial fluctuations, interfaces, defects, and correlation lengths, while reducing to Landau theory for uniform fields.

For a real field, ${\varphi }_{-\mathbf{k}}={\varphi }_{\mathbf{k}}^{\ast }$. The stated Fourier transform givesUsingin both quadratic terms yields

Write $A(k,T)=\gamma k^2+{\mu }^2(T)$ and use units with $k_B=1$. The Gaussian functional integral gives the fluctuation free-energy density, up to terms linear in $T$ that do not affect the heat capacity,Since the heat capacity per volume is $c=-T{d}^{2}f/d{T}^2$,where a dot denotes $d/dT$. ThusFor ${\mu }^2=a(T-T_c)$, these reduce to $g(T)=2aT$ and $h(T)=a^2{T}^2$.

The most singular contribution as ${\mu }^2\downarrow 0$ isRescaling $\mathbf{k}=(\mu /\sqrt{\gamma })\mathbf{q}$ shows that its singular part is proportional toBecause ${\mu }^2\propto T-T_c$, the heat-capacity critical exponent isfor $d<4$. The term with one propagator is less singular. At $d=4$ the power is replaced by a logarithmic singularity, identifying four as the upper critical dimension of this Gaussian heat-capacity correction.

The stationary points obeyNonzero stationary points exist whenso they first appear at the ordered-phase spinodal point $a_2=9a_3^2/(32a_4)$. The disordered state $m=0$ is locally stable for $a_2>0$ and loses that stability at $a_2=0$.

The actual phase boundary is found by requiring a nonzero stationary point $m_{\ast }$ to have the same free energy as $m=0$. Solving $f^{\prime }(m_{\ast })=0$ and $f(m_{\ast })=0$ givesThe disordered state is the global minimum for $a_2>a_3^2/(4a_4)$, the ordered state is the global minimum for $a_2<a_3^2/(4a_4)$, and they coexist at equality.

Since $a_3\ne 0$, the order parameter jumps from zero to $-a_3/(2a_4)$ at coexistence. Hence this model has no continuous phase transition as the phases exchange stability, but it does have a first-order phase transition at the displayed positive value of $a_2$.