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

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 $d\le 2$.