 Thermal eigenvalues of the spherical nonlinear sigma model

With $A_D=S_D{\Lambda }^{ϵ}/(2\pi {)}^D$, the one-loop renormalization-group beta function is ${\beta }_g=-ϵg+(n-2)A_D{g}^2$. For $n>2$ and $ϵ>0$, its two nonnegative renormalization-group fixed points are zero and $g_{\ast }=ϵ/[(n-2)A_D]$. Differentiating the beta function gives the displayed thermal eigenvalues, so the nonzero point has correlation-length critical exponent $\nu =1/ϵ$ to leading order. At $ϵ=0$ the zero eigenvalue requires nonlinear analysis: $g$ is marginally relevant for $n>2$, but the $n=2$ perturbative spin-wave theory has a line of fixed points and omits vortices.