Renormalization-group fixed point

A renormalization-group fixed point is a set of dimensionless couplings unchanged by the stated coarse-graining and rescaling transformation. Its beta functions vanish, and its long-distance description has scale-invariant correlation laws.

Linearizing the renormalization-group flow around $g_{\ast }$ gives $d\delta g_i/d\ell ={\sum }_j{M}_{ij}\delta g_j$. With $\ell$ increasing toward long distances, positive eigenvalues identify perturbations that grow, negative ones identify perturbations that decay, and zero eigenvalues require nonlinear analysis. A physical critical system must tune its relevant perturbations appropriately. The fixed point therefore describes universal critical behavior, while the stability directions describe how to reach or leave it. A zero of a beta function alone does not specify whether it attracts infrared or ultraviolet flows.

An infrared fixed point attracts renormalization-group flows toward low energies or long distances. If it is interacting, it defines a nontrivial scale-invariant quantum or statistical field theory.

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

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