  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.
