Thermal eigenvalues of the spherical nonlinear sigma model ($y_T(g=0)=-ϵ,y_T(g_{\ast })=ϵ$)

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.

At $D=2$ and $n>2$, the weak-coupling renormalization-group flow obeys $dg/d\ell =(n-2)g^2/(2\pi )$. Integrating gives $1/g(\ell )=1/g_0-(n-2)\ell /(2\pi )$. The flow reaches order-one coupling at a length whose logarithm is $2\pi /[(n-2)g_0]+O(1)$ at this order. Restoring the microscopic scale ${\Lambda }^{-1}$ gives the displayed leading exponential correlation length. Higher loops affect its prefactor, and this conclusion does not apply to the compact $n=2$ model with its Berezinskii–Kosterlitz–Thouless transition.