Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 303 (Statistical Field Theory)

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 $d_c=4$.

A Wilsonian renormalization group step first splits the field into slow and fast Fourier modes and integrates out the shell $\Lambda /b<|k|<\Lambda$. It then rescales coordinates, or momenta, to restore the cutoff to $\Lambda$, and finally rescales the field to normalize the kinetic term. The resulting local effective free energy has the same allowed operators with changed coefficients. Repeating the operation therefore composes maps on the set of couplings and defines a renormalization-group flow.

For couplings $g_i(\ell )$ under coarse-graining scale $b=e^{\ell }$, their beta function is ${\beta }_i(g)=d{g}_i/d\ell$. At a renormalization-group fixed point $g_{\ast }$ all beta functions vanish. Linearization givesThe eigenvalues $y_a$ of the stability matrix of a renormalization-group fixed point are the scaling dimensions of the associated coupling directions: $\delta g_a(\ell )=e^{y_a\ell }\delta g_a(0)$.

A coupling direction is relevant, irrelevant, or marginal according as its renormalization-group eigenvalue is positive, negative, or zero. The critical surface is the stable manifold of a critical fixed point: initial couplings on it flow toward that fixed point, while relevant perturbations take the theory away from criticality. Microscopic models with different irrelevant couplings lose those distinctions under coarse-graining and approach the same fixed point, which explains their shared long-distance behavior and universality class.

Split $\varphi ={\varphi }_{<}+{\varphi }_{>}$ and write $I_n={\int }_{\Lambda /b<|k|<\Lambda }\frac{d^{d}k}{(2\pi {)}^d}{G}_0(k{)}^n$. The first cumulant of $g_0{\varphi }^4$ contains $6g_0{\varphi }_{<}^2⟨{\varphi }_{>}^2⟩=6g_0{I}_1{\varphi }_{<}^2$. The connected second cumulant of the two terms $3{\lambda }_0{\varphi }_{<}{\varphi }_{>}^2$ uses Wick theorem to give $⟨{\varphi }_{>}^2(x){\varphi }_{>}^2(y){⟩}_c=2G_{>}(x-y{)}^2$. Keeping its leading local term, the quadratic coefficient after integrating out the shell is thereforeThe quartic tadpole raises the mass parameter, while the pair of cubic vertices gives the negative bubble contribution. A cubic interaction also generates a linear tadpole $3{\lambda }_0{I}_1{\varphi }_{<}$, which may be removed by shifting the background but is not part of ${\mu }^{\prime 2}$.

The first correction to the four-point coupling containing ${\lambda }_0$ is of order ${\lambda }_0^2$: two cubic vertices are joined by one fast internal propagator, leaving four slow external legs. The three exchange channels pair the external legs in the $s$, $t$, and $u$ ways. In a sharp momentum-shell scheme this contribution is retained when the exchanged momentum lies in the eliminated shell; its derivative expansion supplies the induced local quartic interaction.

If ${\lambda }_0=0$, the action has the exact discrete symmetry $\varphi \mapsto -\varphi$. Integrating out modes and rescaling preserve this symmetry, so no odd operator such as ${\varphi }^3$ can be generated from the even quartic coupling. Thus $g_0$ corrects $\lambda$ at no order in perturbation theory: the hypersurface $\lambda =0$ is invariant under the renormalization-group flow.

If the singular free-energy density is set by one correlated region, then $f_s\sim {\xi }^{-d}$. With reduced temperature $t$, the definitions $\xi \sim |t{|}^{-\nu }$ and $f_s\sim |t{|}^{2-\alpha }$ therefore giveThis is the hyperscaling relation; it assumes that no dangerously irrelevant coupling introduces an additional scale.

For ${\mu }^2<0$ and $g>0$, minimizing the $O(N)$-invariant potential givesThe free energy has the full $O(N)$ symmetry, but choosing one point on this sphere leaves only the rotations $O(N-1)$ fixing that point. This is spontaneous symmetry breaking, $O(N)\to O(N-1)$. The $N-1$ tangent directions along the sphere cost no potential energy and are the massless Goldstone modes predicted by the Goldstone theorem.

Choose the vacuum $\varphi =(v+\sigma ,{\pi }_1,\dots ,{\pi }_{N-1})$. The quadratic free energy contains a positive mass term for the radial field $\sigma$ but no mass term for any transverse field ${\pi }_a$:Thus the momentum-space correlation function is $⟨{\pi }_a(k){\pi }_b(-k)⟩={\delta }_{ab}/k^2$. Since a massive correlator has denominator $k^2+{\xi }^{-2}$, these Goldstone modes have ${\xi }^{-1}=0$ and hence infinite correlation length.

The long-wavelength Goldstone fluctuation is proportional towhich diverges in the infrared for $d\le 2$. These fluctuations destroy finite-temperature long-range order with a broken continuous symmetry, as formalized by the Mermin-Wagner theorem. Therefore the lower critical dimension is $d_l=2$: conventional spontaneous $O(N)$ order is possible for $d>2$ but not for $d\le 2$.

The $O(2)$ rotation symmetry requires the quadratic dependence on $({\varphi }_2,{\varphi }_3)$ to be proportional to ${\varphi }_2^2+{\varphi }_3^2$, soThe quartic terms involving only that pair must be proportional to $({\varphi }_2^2+{\varphi }_3^2{)}^2$, and the terms coupling it to ${\varphi }_1$ must be proportional to ${\varphi }_1^2({\varphi }_2^2+{\varphi }_3^2)$. With the convention that the symmetric double sum counts off-diagonal terms twice, this giveswhile ${\mu }_1^2$, $g_{11}$, and the common mixed coupling are unrestricted. The stated ${\mathbb{Z}}_2$ symmetry imposes no further relation because every term is already even in ${\varphi }_1$.

Let $r_1={\mu }_1^2$, $r_2={\mu }_2^2={\mu }_3^2$, and ${\rho }^2={\varphi }_2^2+{\varphi }_3^2$. With every $g_{ij}=g>0$, the uniform potential isFor $r_1>0,r_2>0$, the minimum is the origin and ${\mathbb{Z}}_2\times O(2)$ is unbroken. If $r_1<\min (0,r_2)$, then ${\varphi }_1^2=-r_1/(4g)$ and $\rho =0$; the ${\mathbb{Z}}_2$ factor is broken, $O(2)$ remains, and there is no Goldstone mode. If $r_2<\min (0,r_1)$, then ${\varphi }_1=0$ and ${\rho }^2=-r_2/(4g)$; the first ${\mathbb{Z}}_2$ remains while $O(2)$ is broken to the reflection fixing the chosen direction, producing one Goldstone mode.

The positive coordinate half-axes are continuous transition lines. For $r_1=r_2<0$, the potential has an enhanced $O(3)$ symmetry and a sphere of minima; $O(3)\to O(2)$ gives two Goldstone modes on this line. Crossing the negative diagonal exchanges the two ordered phases and gives a first-order line at mean-field level.