= Thermal eigenvalues of the spherical nonlinear sigma model
{title2=$y_T(g=0)=-\epsilon,\quad y_T(g_*)=\epsilon$}
With $A_D=S_D\Lambda^\epsilon/(2\pi)^D$, the one-loop <renormalization-group beta function> is $\beta_g=-\epsilon g+(n-2)A_Dg^2$. For $n>2$ and $\epsilon>0$, its two nonnegative <renormalization-group fixed points> are zero and $g_*=\epsilon/[(n-2)A_D]$. Differentiating the beta function gives the displayed thermal <eigenvalues>, so the nonzero point has <correlation-length critical exponent> $\nu=1/\epsilon$ to leading order. At $\epsilon=0$ 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.
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