theory, and not just perturbative data.

To resolve this ambiguity, we turn to an approach which is equivalent to the former one at the [*scaling*]{} level, but which is more quantitative. A natural choice is the Chain Mean Field Theory (CMFT), in which inter-chain couplings are treated by a self-consistent Weiss-type decoupling, using the exact solutions of perturbed but decoupled individual chain problems [@schulz-cmft]. Our CMFT approach is described in detail in Appendix \[sec:cmft\]. In principle, it can be employed to determine a full mean-field phase diagram. Here, we will mostly use it in more limited ways, as convenient. To address the ambiguity discussed above, we use the CMFT to compute a putative ordering temperature, $T_i$, for each channel driven by an operator $\mathcal{O}_i$. The instability which sets in first upon lowering the temperature, i.e. with maximal $T_i$, is assumed to be dominant. Another application of CMFT will be to compute the magnitude of the ordering induced by a coupling $\gamma_i$, at zero temperature. This will be useful in making quantitative estimates of more subtle smaller energy scales, as we will see below.

Ideal 2d model {#sec:ideal-2d-model}
==============

In this section, we consider the behavior of the ideal 2d model, 