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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 $\xi\sim|t|^{-\nu}$.

Correlation length

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

Canonical quantization imposes
$$
[a_{\mathbf k},a_{\mathbf k'}^\dagger]=(2\pi)^3\delta^{(3)}(\mathbf k-\mathbf k'),
\qquad [a_{\mathbf k},a_{\mathbf k'}]=[a_{\mathbf k}^\dagger,a_{\mathbf k'}^\dagger]=0.
$$
The <correlation function>[two-point correlation function] has mode power $P_{\delta\phi}=|f_k|^2/a^2$. After <cosmological horizon exit>, $|k\tau|\ll1$, so
$$
|f_k|^2\simeq\frac1{2k^3\tau^2},
\qquad a^2=\frac1{H^2\tau^2}.
$$
Therefore
$$
\Delta_{\delta\phi}^2=\frac{k^3}{2\pi^2}P_{\delta\phi}
=\boxed{\left(\frac H{2\pi}\right)^2}.
$$
This is the <scale-invariant inflationary power spectrum> of a light canonical scalar.

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 310 / 4 / a / Solution

The eliminated field obeys $\phi=-cp^2/(2a)$, so its connected <correlation function> is
$$
C_\phi(r)=\frac{c^2}{4a^2}\left[\langle p^2(0)p^2(r)\rangle-\langle p^2\rangle^2\right].
$$
For the zero-mean Gaussian field of part (i), <Wick theorem> gives
$$
\boxed{C_\phi(r)=\frac{c^2}{2a^2}C_p(r)^2}.
$$
It is therefore nonnegative and has the square of the Ornstein--Zernike spatial form. In particular, if $C_p(r)$ has exponential factor $e^{-r/\xi_p}$, then $C_\phi(r)$ has $e^{-2r/\xi_p}$ and correlation length $\xi_p/2$, with the algebraic prefactor also squared.

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 344 / 1 / e / ii / Solution

{wiki}

A correlation function measures the joint fluctuation of observables at separated points. For a scalar field, the connected two-point function is $G(x-y)=\langle\phi(x)\phi(y)\rangle-\langle\phi(x)\rangle\langle\phi(y)\rangle$.


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