described by the standard model in Eqs. (\[eq:Hexpt\],\[eq:1\]) with $J''=D=0$.

Continuum limit {#sec:continuum-limit}
---------------

For this case, the technology of the previous section can be straightforwardly applied. We begin with the naïve procedure of simply inserting the decompositions in Eq. (\[eq:1dspindecomp\]) into the microscopic inter-chain lattice Hamiltonian in Eq. (\[eq:1\]) (We will assess the need to go beyond this approximation later). Specifically, we have $$\begin{aligned}
  \label{eq:2}
   S_{x,y,z}^z & \sim &  M + \mathcal{S}^z_{y,z;0}(x) \\
&& + e^{i(\pi-2\delta)x} \mathcal{S}^z_{y,z;\pi-2\delta}(x) + e^{-i(\pi-2\delta)x} \mathcal{S}^z_{y,z;\pi+2\delta}(x) , \nonumber \\
S_{x,y,z}^+& \sim &  e^{-i2\delta x} \mathcal{S}^+_{y,z;2\delta}(x)\nonumber \\
&& 
    + e^{i 2\delta x} \mathcal{S}^+_{y,z;-2\delta}(x) + e^{i\pi x}
  \mathcal{S}^+_{y,z;\pi}(x). \nonumber\end{aligned}$$ It might appear that a slightly different formula should be applied for odd chains, since in those cases in our convention the $x$ coordinates are half-integer rather than integer. However, the differences can be removed by constant shifts of $\theta$ and $\phi$ for the odd chains, without any further changes. Hence we can uniformly apply Eq. (\[eq:2\]) to all chains.

Having dropped the $D$ term, the behavior of the model at zero temperature is independent of the direction of the field, and is a function only of the magnetization $M$ and the magnitude of $J'$. Inserting the decomposition 