Spectral gap in mean-field $\mathcal{O}(n)$-model

We study the dependence of the spectral gap for the generator of the Ginzburg-Landau dynamics for all \emph{$\mathcal{O}(n)$-models} with mean-field interaction and magnetic field, below and at the critical temperature on the number $N$ of particles. For our analysis of the Gibbs measure, we use a one-step renormalization approach and semiclassical methods to study the eigenvalue-spacing of an auxiliary Schr"odinger operator.