Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 337 / 1 / b

Every term in $\tilde{S}$ is proportional to $N$. The Large-N expansion therefore suppresses fluctuations of the auxiliary field by powers of $1/N$, and the leading partition function comes from a saddle-point approximation.

Vary with respect to $i\lambda$ and use $\delta \mathrm{Tr}\log K=\mathrm{Tr}(K^{-1}\delta K)$. At a translation-invariant saddle $i\lambda =m^2$, the gap equation isThe frequencies are the bosonic Matsubara frequencies imposed by periodicity around the imaginary-time thermal circle.