Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 303 / 1

For the partition function $Z={\sum }_{\{S\}}{e}^{-\beta E}$, differentiation with respect to the external field givesTherefore the magnetization per site is

The mean-field approximation replacesby its first three terms. Since every site has $q$ neighbours,The sites then decouple, and their three possible weights sum toHence $c_1=1$, $c_2=2$, and

Using the field derivative while imposing the self-consistency equation givesAt $B=0$, $m=0$ is always a solution. Linearizing the right-hand side for small $m$ gives slopeA continuous phase transition occurs when this slope crosses one, so, with $k_B=1$ and ${\kappa }_c=e^{-g/T_c}$,On the side where the slope exceeds one, the nonzero solutions lower the mean-field free energy and the trivial solution is unstable. The equation has a physical continuous-transition solution only in the corresponding parameter range of this spin-one model.

Landau-Ginzburg theory assumes a local gradient expansion for a slowly varying order parameter. At zero field the microscopic spin-flip symmetry sends $m\mapsto -m$, so only even powers occur. Rotational symmetry makes the leading derivative term $(\nabla m{)}^2$, while locality and analyticity near the transition organize higher powers and derivatives. Stability requires a positive gradient coefficient and, at an ordinary continuous transition, ${\alpha }_4(T_c)>0$; the coefficient ${\alpha }_2(T)$ changes sign at $T_c$.

Write $t=T-T_c$, take ${\alpha }_2=at$ with $a>0$, and include the magnetic term $-Bm$. At $B=0$, minimizing the Landau free energy gives $m=0$ for $t>0$ andfor $t<0$, so the order-parameter critical exponent is $\beta =1/2$. Substitution gives a singular free-energy density proportional to $-t^2$ below $T_c$, whose second temperature derivative has a finite jump, so the heat-capacity exponent is $\alpha =0$. Above $T_c$, the equation ${\alpha }_{2}m+{\alpha }_4{m}^3=B$ gives the magnetic susceptibility $\chi =(\partial m/\partial B{)}_{B=0}=1/{\alpha }_2\sim t^{-1}$, hence $\gamma =1$. At $t=0$, $B={\alpha }_4{m}^3$, so $m\sim B^{1/3}$ and $\delta =3$.

The Ginzburg criterion compares fluctuations of the order parameter averaged over one correlation volume with the squared mean-field order parameter. The stated correlation function givesMeanwhile $m_0^2\sim |t|\sim {\xi }^{-2}$, so their ratio scales as ${\xi }^{4-d}$. It vanishes at criticality for $d>4$, grows for $d<4$, and is marginal at $d=4$. Thus the ordinary scalar Landau-Ginzburg model has upper critical dimension $d_c=4$.