Renormalization-group relevance

The relevance of a perturbation is determined by the eigenvalue of its dimensionless coupling under linearized renormalization-group flow. Positive, negative and zero eigenvalues define relevant, irrelevant and marginal perturbations, respectively.

An operator is relevant at a fixed point when its coupling has positive renormalization-group eigenvalue. Its dimensionless coupling grows under coarse-graining and drives the theory away from the fixed point unless tuned.

A perturbation of a renormalization-group fixed point is relevant if it grows as the coarse-graining length increases. For a continuous flow $d\delta g/d\ell =M\delta g$, an eigenvector of the linearized matrix with eigenvalue $\sigma$ grows as $e^{\sigma \ell }$; a positive real part is relevant, a negative real part is irrelevant and a zero real part requires nonlinear analysis. For a discrete transformation with length factor $b>1$, its Jacobian matrix multiplier $\lambda$ instead gives $\delta g_n={\lambda }^n\delta g_0$: $|\lambda |>1$ is relevant, $|\lambda |<1$ is irrelevant and $|\lambda |=1$ requires nonlinear analysis. For a positive multiplier the scaling exponent is $y=\log \lambda /\log b$. Thus a positive discrete eigenvalue smaller than one is an irrelevant operator direction, while the sign criterion belongs to the continuous generator.

A discrete renormalization group with length factor $b>1$ and positive relevant thermal multiplier ${\lambda }_t>1$ sends $t$ to ${\lambda }_{t}t$ and the correlation length to $\xi /b$. Combining this with $\xi \sim |t{|}^{-\nu }$ gives the displayed correlation-length critical exponent. The corresponding scaling dimension is $y_t=\log {\lambda }_t/\log b=1/\nu$. Relevance of discrete multipliers is determined by magnitude relative to one, unlike the sign criterion for continuous-flow eigenvalues.

An operator is irrelevant at a fixed point when its coupling has negative renormalization-group eigenvalue. Its dimensionless coupling decreases under coarse-graining.

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.