Irrelevant variables

We may also include the irrelevant variables in the scaling law of :

where and . After iterations we have:
where in accordance with the fact that . Setting we have:
For the terms involving the irrelevant variables become vanishingly small, so we get:
Note that in the last step we have implicitly assumed that is analytic in the limit . This assumption is however frequently false! When this happens, i.e. when the free energy density is singular in the limit for a particular irrelevant variable , that variable is termed dangerous irrelevant variable. For example, considering the Landau free energy of the Ising model obtained as a saddle-point approximation of the general functional partition function:
the parameter of the quartic term is a dangerous irrelevant variable (we have seen in Coarse graining procedure for the Ising model that problems arise when we try to treat it as a perturbative parameter).