Choose the vacuum $\phi=(v+\sigma,\pi_1,\ldots,\pi_{N-1})$. The quadratic free energy contains a positive mass term for the radial field $\sigma$ but no mass term for any transverse field $\pi_a$:
$$
F^{(2)}_\pi=\frac12\sum_{a=1}^{N-1}\int d^dx,(\nabla\pi_a)^2.
$$
Thus the momentum-space <correlation function> is $\langle\pi_a(k)\pi_b(-k)\rangle=\delta_{ab}/k^2$. Since a massive correlator has denominator $k^2+\xi^{-2}$, these Goldstone modes have $\xi^{-1}=0$ and hence infinite <correlation length>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 303 / 3 / b / ii / Solution

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A phase transition is a nonanalytic change in equilibrium behavior as a thermodynamic control parameter varies. A continuous phase transition has a diverging <correlation length> and no latent heat.

Phase transition

{title2=$\xi$}
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The correlation length is the characteristic distance over which a connected <correlation function> decays. It diverges at an ordinary continuous critical point as $\xi\sim|t|^{-\nu}$.

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 Incoming links: Correlation length