= Marginally irrelevant operator
= Marginally irrelevant
{synonym}
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 is
$$
\frac{dg}{d\ell}=-ag^2+O(g^3),\qquad a>0,\quad g>0.
$$
To leading order, $g(\ell)=g_0/(1+ag_0\ell)$, 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.
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