Marginal operator

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.

A marginally irrelevant operator has a coupling that is unchanged to linear order by the renormalization group, but flows slowly toward zero once nonlinear terms are included. For an increasing coarse-graining length with logarithmic scale $\ell$, a simple example isTo leading order, $g(\ell )=g_0/(1+a{g}_0\ell )$, so the perturbation decays as the inverse logarithm of length rather than as a power of length.

The coupling leaves the limiting universality class unchanged, but can produce logarithmic corrections and slow convergence in finite systems. Ignoring it can make an apparent critical exponent drift with the observation scale. The direction of the flow and the sign of the initial coupling matter: the same local formula sends negative $g$ away from zero rather than making every perturbation irrelevant.