of Eq. (\[eq:1dspindecomp\]) into $H_1$ in Eq. (\[eq:1\]), one need keep only terms which do not oscillate, the condition corresponding to momentum conservation. Using the slowly-varying nature of the scaling operators in $x$ (but [*not*]{} in $y$ and/or $z$), one may take a continuum limit in $x$ by gradient expansion to obtain the lowest non-vanishing terms of each type. One finds:

$$\begin{aligned}
  \label{eq:H1dec}
  H_1 & \approx & J' \sum_{y,z} \int\! dx\, \Big\{ 2 M^2 + 2
  \mathcal{S}_{y,z;0}^z\mathcal{S}_{y+1,z;0}^z +
  2\sin \delta\left[ \mathcal{S}_{y,z;\pi-2\delta}^z 
  \mathcal{S}_{y+1,z;\pi+2\delta}^z+ {\rm h.c.}\right] \nonumber \\
  & & +\frac{1}{2} \left[-i\mathcal{S}_{y,z;\pi}^+ \partial_x
  \mathcal{S}_{y+1,z;\pi}^- + {\rm h.c.}\right] + \cos\delta \left[
   \mathcal{S}_{y,z;2\delta}^+
    \mathcal{S}_{y+1,z;2\delta}^- + \mathcal{S}_{y,z;-2\delta}^+
    \mathcal{S}_{y+1,z;-2\delta}^- + {\rm h.c.}\right] \Big\}\end{aligned}$$

Let us now assess the importance of each of the terms in Eq. (\[eq:H1dec\]). This is accomplished by ranking each of the terms in order of increasing scaling dimension, or equivalently, decreasing relevance in the RG sense. Formally, the most relevant term is the first ($M^2$) one, which is a $c$-number constant and hence of dimension zero. Though it is a “trivial” constant (at fixed magnetization) and hence does not affect the dynamics of the system, it is indeed the dominant correction to the ground state energy of the weakly coupled chains. Being 