A Wilsonian renormalization-group step integrates out short-distance modes, rescales coordinates to restore the cutoff, and rescales fields to normalize a chosen kinetic term.
At the Gaussian fixed point all interaction couplings vanish and scaling dimensions follow from the quadratic theory.
Below four dimensions, the scalar quartic theory has an interacting Wilson-Fisher fixed point that controls the Ising universality class.
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.
Systems belong to one universality class when their long-distance renormalization-group flows approach the same fixed point.
An irrelevant coupling is dangerously irrelevant when setting it to zero makes a scaling function singular or removes the term needed to stabilize the ordered phase.
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The renormalization group (RG) is a mathematical and conceptual framework used in theoretical physics to study changes in a physical system as one looks at it at different scales. It is particularly prominent in quantum field theory, statistical mechanics, and condensed matter physics. The central idea behind the RG is that the properties of a system can change when one changes the scale at which one observes it.
