Renormalization Group

A renormalization group is a family of transformations that changes the scale at which a model is described while preserving its long-distance behavior. Let denote the model's coupling parameters and let be the transformation associated with a scale factor . Successive changes of scale satisfy

(1)

Because coarse-graining discards short-distance information, need not be invertible, so the transformations generally form a semigroup rather than a group.

A fixed point satisfies

(2)

Near such a point, the transformation can be linearized as

(3)

where the eigenvalues of are conventionally written . A direction is called relevant, irrelevant, or marginal according as is positive, negative, or zero. Relevant perturbations grow under coarse-graining, irrelevant perturbations decay, and marginal perturbations require higher-order analysis.

Writing gives the continuous form of the renormalization-group flow,

(4)

where the are called beta functions and their simultaneous zeros are fixed points. At a continuous phase transition, a fixed point with one relevant thermal scaling field of exponent gives the correlation length critical exponent

(5)

Different microscopic models flowing to the same fixed point therefore have the same long-distance scaling behavior and critical exponents (Wilson 1971, Wilson and Kogut 1974, Cardy 1996).