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The correlation-length critical exponent is defined by $\xi \sim |t{|}^{-\nu }$. At a renormalization-group fixed point, its reciprocal is the positive eigenvalue of the temperature-like scaling direction.

The critical surface is the stable manifold of a critical renormalization-group fixed point. Tuning every relevant direction places a theory on this surface.

The engineering value follows from the Gaussian kinetic term. At an interacting renormalization-group fixed point, momentum-dependent self-energy diagrams change the kinetic coefficient, and restoring its normalization requires wave-function renormalization. The resulting anomalous dimension $\eta$ changes the full scaling dimension to

For couplings $g_i(\ell )$ under coarse-graining scale $b=e^{\ell }$, their beta function is ${\beta }_i(g)=d{g}_i/d\ell$. At a renormalization-group fixed point $g_{\ast }$ all beta functions vanish. Linearization givesThe eigenvalues $y_a$ of the stability matrix of a renormalization-group fixed point are the scaling dimensions of the associated coupling directions: $\delta g_a(\ell )=e^{y_a\ell }\delta g_a(0)$.

The stability matrix is the Jacobian of the beta functions at a renormalization-group fixed point. Its eigenvalues classify perturbations as relevant, irrelevant, or marginal.