Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 337 / 1 / d / Solution

For bosonic Matsubara frequencies ${\omega }_n=2\pi nT$, set $a=vk/(2\pi T)$. Applying the residue theorem to $\pi \coth (\pi z)/(z^2+a^2)$, whose integer poles reproduce the desired summands, givesIt follows thatUsing the area $S_{d-1}=2{\pi }^{d/2}/\Gamma (d/2)$ of the unit sphere, the thermal phase fluctuation becomesAt small $k$, $\coth (vk/(2T))\sim 2T/(vk)$, so the infrared divergence is governed by ${\int }_{0}dk{k}^{d-3}$. It diverges for $d=1,2$ and is finite for $d=3$. Therefore short-range systems cannot have true finite-temperature breaking of this continuous symmetry in one or two dimensions, in agreement with the Mermin-Wagner theorem, whereas it is allowed in three dimensions. In two dimensions a Berezinskii–Kosterlitz–Thouless transition may still produce quasi-long-range order.