implies $\Delta<2$ for that coupling. The RG is valid only so long as the [*largest*]{} of these dimensionless couplings remains small. Crudely, then, we may determine the length scale $\xi$ at which inter-chain coupling becomes significant by the point $\ell=\ell^*$ [*at which the first operator renormalizes to become of $O(1)$*]{}, where $\xi=a e^{\ell^*}$. The length $\xi$ defines a correlation length, below which the dynamics is approximately one-dimensional.

If this first “diverging” operator is unique, one can often identify the nature of the associated instability of the decoupled chains. In many cases, this can be done by dropping the other operators, and treating the remaining one by a type of mean field theory. Sometimes it can be treated in a semiclassical fashion. None of these approaches are rigorous, but they are eminently reasonable, and are likely to correctly predict the nature of the resulting state.

$ $From this reasoning, we see that both the bare magnitude and the scaling dimension (relevance) of the different interactions are important in determining the low energy state of the system. To be more concrete, consider two candidate operators, $\mathcal{O}_1,\mathcal{O}_2$. Their renormalized coupling constants obey $$\label{eq:32}
  \breve{\gamma}_i(\xi) = \frac{\gamma_i}{v} \xi^{2-\Delta_i}.$$ Here, since $\Lambda=\Lambda_{\ell=0}$ is $O(1)$, we 