jats:titleAbstract</jats:title>jats:pWe study the dependence of the spectral gap for the generator of the Ginzburg–Landau dynamics for alljats:inline-formulajats:alternativesjats:tex-math$$\mathcal O(n)$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">mml:mrowmml:miO</mml:mi>mml:mo(</mml:mo>mml:min</mml:mi>mml:mo)</mml:mo></mml:mrow></mml:math></jats:alternatives></jats:inline-formula>jats:italic-models</jats:italic>with mean-field interaction and magnetic field, below and at the critical temperature on the numberjats:italicN</jats:italic>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 Schrödinger operator.</jats:p>