Renormalization-group flow

A renormalization-group flow tracks how a theory's effective couplings change as the observation scale changes. A coarse-graining step averages short-distance degrees of freedom and rescales the remaining variables, producing a new description for longer-distance measurements.

For dimensionless couplings $g_i$ and a logarithmic length scale $\ell$, write $d{g}_i/d\ell ={\beta }_i(g)$. A renormalization-group fixed point has all beta functions zero. Near it, some perturbations grow and others shrink, giving relevant operators and irrelevant operators. This explains how distinct microscopic models can approach the same long-distance behavior. The sign convention matters: a beta function defined using increasing energy has the opposite scale direction from one defined using increasing length.

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 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.