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For a fixed point $g_f$, put $g=g_f+\delta g$. Linearizing the renormalization-group flow givesBecause $g$ is proportional to temperature at fixed microscopic stiffness, this derivative is the thermal eigenvalue. The thermal eigenvalues of the spherical nonlinear sigma model areThe first value describes the stable zero-temperature ordered fixed point. At the nonzero critical point the thermal perturbation grows as $e^{ϵ\ell }$, so the stopping scale gives $\xi \propto |g-g_{\ast }{|}^{-1/ϵ}$ and $\nu =1/ϵ$ to leading order in the expansion. Higher-loop terms change this to $y_T=ϵ+O({ϵ}^2)$.

At $ϵ=0$ the linear eigenvalue is zero, so its sign alone does not decide the flow. For $n>2$ the first nonlinear term is positive. Integrating it givesThe flow reaches order-one coupling at a length logarithm of order $2\pi /[(n-2)g(0)]$, explaining the exponentially large correlation length in a two-dimensional spherical sigma model rather than a finite-temperature power-law critical point in this smooth-field regime. For $n=2$, $ϵ=0$, the linear eigenvalue is zero everywhere on the perturbative line and vortex effects require an additional coupling. For $n=2$, $ϵ>0$, the sole zero-coupling fixed point has $y_T=-ϵ$.