Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 303 (Statistical Field Theory)

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 $a_2$.

A momentum-shell renormalization group step has three parts:

1. Split the field into slow modes ${\varphi }^{-}$ with $|k|<\Lambda /\zeta$ and fast modes ${\varphi }^{+}$ with $\Lambda /\zeta <|k|<\Lambda$, then integrate out ${\varphi }^{+}$ to obtain a Wilsonian effective action for ${\varphi }^{-}$.
2. Rescale momenta by $k^{\prime }=\zeta k$, equivalently coordinates by $x^{\prime }=x/\zeta$, so the reduced cutoff returns from $\Lambda /\zeta$ to $\Lambda$.
3. Rescale the field, at the Gaussian fixed point by ${\varphi }^{\prime }(x^{\prime })={\zeta }^{(d-2)/2}{\varphi }^{-}(x)$, so the coefficient of $(\nabla \varphi {)}^2/2$ retains its chosen normalization.

The resulting functional has the original cutoff but changed coefficients. Iterating the operation gives a renormalization-group flow on masses and interaction couplings.

At the Gaussian fixed point, the engineering dimension of the field isThe operator ${\varphi }^n({\nabla }^2\varphi {)}^m$ contains $n+m$ fields and $2m$ derivatives. Since its integral must be dimensionless,It is marginal when this vanishes, namelywhen $n+m\ne 2$. Since $n,m$ are positive, the exceptional case is $n=m=1$: $\varphi {\nabla }^2\varphi$ has a dimensionless coupling in every $d$ and differs from the kinetic term by integration by parts. If the displayed formula gives no positive $d$, there is no positive spatial dimension in which that operator is naively marginal.

At an interacting fixed point, field and composite operators acquire anomalous dimensions, and operators with the same symmetries can mix under renormalization. Their full scaling dimensions can therefore differ from these naive engineering dimensions.

Use two external slow-field legs and internal fast-mode propagators. Through the requested orders, the connected mass-correction topologies are:

A nominal one-line bridge at order ${\lambda }_0{g}_0$ vanishes in a sharp momentum-shell scheme at small external momentum because that line would have to carry momentum outside the fast shell. These descriptions specify the same diagrams without depending on a particular drawing convention for vertices and external legs.

Writeand, for the three-line topology,and define the four-line integralwith every propagator momentum restricted to the fast shell.

The first cumulant containsSince the quadratic free energy is $\frac{1}{2}{\mu }^2({\varphi }^{-}{)}^2$, this gives

The mixed term in the second cumulant isWick contraction with the printed normalization $V_6={\lambda }_0\int {\varphi }^6$ and $V_4=g_0\int {\varphi }^4$ gives, at zero external momentum,The coefficients respectively combine the two placements of both external legs in the two-line topology, the three-line topology with one external leg on each vertex, and the four-line topology. Couplings normalized as $g_0{\varphi }^4/4!$ and ${\lambda }_0{\varphi }^6/6!$ absorb the corresponding factorials, which is why formulas in that convention have much smaller numerical coefficients.

For a uniform field with $r^2=\mathbf{\varphi }\cdot \mathbf{\varphi }$, the potential isFor $T>T_c$, one has ${\mu }^2>0$, so ${\mathbf{\varphi }}_0=0$. The Hessian has $N$ positive equal eigenvalues, and all $N$ modes are gapped. The full $O(N)$ symmetry is unbroken.

For $T<T_c$, one has ${\mu }^2<0$, andThe vacuum manifold is $S^{N-1}$. Choosing one point breaks $O(N)$ spontaneously to $O(N-1)$. The radial fluctuation has squared mass $-2{\mu }^2$, while the $N-1$ tangent directions are gapless Nambu-Goldstone bosons, one for each broken continuous generator modulo the unbroken subgroup.

For the $O(2)$ model writeAt long distances the massive radial field $\rho$ can be neglected, leaving the Goldstone-mode effective free energyThereforeThe mode is massless, so its correlation length is infinite. For $d>2$ its large-distance Green function isFor $d=2$ it is $-(2\pi \gamma v^2{)}^{-1}\log (r/a)$ up to an infrared-dependent constant, and in $d=1$ it is $-|r|/(2\gamma v^2)$ up to such a constant.

The invariant diagnostic is the phase-difference varianceIt diverges linearly in $d=1$ and logarithmically in $d=2$, destroying true long-range order, but approaches a finite infrared limit for $d=3,4$. Thus the continuous-symmetry ordered phase exists for $d=3,4$ and not for $d=1,2$, in agreement with the Mermin-Wagner theorem. The lower critical dimension is $d_{\mathrm{lc}}=2$; the two-dimensional $O(2)$ model can instead show quasi-long-range order below a BKT transition.

For generic ${\mu }^2\ne {\lambda }^2$, the symmetry is $O(1)\times O(2)$: an independent sign flip of ${\varphi }_1$ and orthogonal transformations of $({\varphi }_2,{\varphi }_3)$. When ${\mu }^2={\lambda }^2$, it is enhanced to $O(3)$.

If ${\mu }^2,{\lambda }^2>0$, the unique ground state is ${\mathbf{\varphi }}_0=0$. The full $O(1)\times O(2)$ symmetry remains unbroken and every mode is gapped.

If ${\mu }^2<0$ and ${\lambda }^2>0$, the minima areThe discrete $O(1)\cong {\mathbb{Z}}_2$ symmetry is spontaneously broken, while $O(2)$ remains intact. There is no Goldstone mode because only a discrete symmetry is broken.

If ${\mu }^2>0$ and ${\lambda }^2<0$, the minima areThe $O(1)$ factor remains unbroken, while $O(2)$ is spontaneously broken to the $O(1)$ subgroup that fixes a chosen point on the circle. The one-dimensional vacuum circle gives one Goldstone mode.

If ${\mu }^2={\lambda }^2<0$, the enhanced $O(3)$-symmetric potential has the sphere of minimaChoosing a ground state breaks $O(3)$ to $O(2)$. The vacuum manifold $O(3)/O(2)\simeq S^2$ has dimension two, so there are two Goldstone modes.