Renormalization-group relevance

The relevance of a perturbation is determined by the eigenvalue of its dimensionless coupling under linearized renormalization-group flow. Positive, negative and zero eigenvalues define relevant, irrelevant and marginal perturbations, respectively.

An operator is relevant at a fixed point when its coupling has positive renormalization-group eigenvalue. Its dimensionless coupling grows under coarse-graining and drives the theory away from the fixed point unless tuned.

A linearized renormalization-group eigenvector with positive eigenvalue grows under coarse-graining and is relevant; a negative eigenvalue is irrelevant and a zero eigenvalue is marginal.

An operator is irrelevant at a fixed point when its coupling has negative renormalization-group eigenvalue. Its dimensionless coupling decreases under coarse-graining.

An operator is marginal at linear order when its coupling has zero renormalization-group eigenvalue. Nonlinear terms in its renormalization-group beta function determine whether it is marginally relevant, marginally irrelevant, or exactly marginal.