Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 303 / 3

The constraint $\mathbf{n}\cdot \mathbf{n}=1$ makes $\mathbf{n}$ dimensionless. Since $d^{d}x({\partial }_i{n}_A{)}^2$ has engineering dimension $2-d$, dimensional consistency of the O(N) nonlinear sigma model gives

Choose the positive local branch of the constraint,The chain rule givesand thereforeSubstitution into the original free energy gives exactly

For small $\mathbf{\pi }$, expand $(1-{\mathbf{\pi }}^2{)}^{-1}=1+O({\mathbf{\pi }}^2)$ and retain the quartic interaction $(\mathbf{\pi }\cdot {\partial }_i\mathbf{\pi }{)}^2/(2g_0)$. Split $\mathbf{\pi }={\mathbf{\pi }}^{-}+{\mathbf{\pi }}^{+}$ and contract the fast fields in the shell $\Lambda /\zeta <|q|<\Lambda$. To first order inthe contraction adds $I_d$ to the inverse coefficient of the slow-field kinetic term. The prescribed wave-function renormalizationthen multiplies that coefficient by $A^2$. HenceFinally the coordinate rescaling contributes ${\zeta }^{d-2}$, and thus

Let $s=\log \zeta$. For an infinitesimal momentum shell,Differentiating the inverse-coupling relation and using $d(1/g)/ds=-g^{-2}dg/ds$ gives the one-loop beta functionFor $d=2+ϵ$, ${\Omega }_{d-1}/(2\pi {)}^d=1/(2\pi )+O(ϵ)$, so

The renormalization-group fixed points are the zeros of the beta function:For $N>2$ and $ϵ>0$, the positive fixed point $g_c^{\ast }$ separates the low-temperature ordered flow toward $g=0$ from the high-temperature strong-coupling flow. At $ϵ=0$ the two perturbative fixed points merge at zero.

Identifying $g$ with temperature, write $g=g_c^{\ast }+\delta g$ near the nonzero critical fixed point. The derivative of the beta function there isThus the temperature-like perturbation has renormalization-group eigenvalue $y_t=ϵ$. Since the correlation-length critical exponent satisfies $\nu =1/y_t$,The Gaussian fixed point $g=0$ is the stable ordered-phase fixed point for $d>2$, rather than the finite-temperature transition. In exactly two dimensions the flow instead gives an essential, exponential correlation-length divergence, corresponding formally to $\nu =\infty$ rather than a finite power-law exponent.