Renormalization-group flow

A renormalization-group flow is the trajectory of effective couplings as the observation scale changes.

A repulsive renormalization-group trajectory leaves a renormalization-group fixed point along its unstable manifold as the coarse-graining length increases. Its tangent directions are relevant operators. The critical surface is instead the stable manifold, whose irrelevant coordinates approach the critical fixed point after relevant fields are tuned. For a discrete step, relevance is determined by multipliers larger than one in magnitude, not merely by their being positive.

A renormalization-group fixed point is a set of couplings left unchanged by coarse-graining. It describes a scale-invariant theory.

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 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.