H_1 + H_2,$$ with $$\begin{aligned}
  \label{eq:1}
  H_1 & = & \sum_{xyz}
J' {\bm S}_{x,y,z}\cdot\left({\bm S}_{x-\frac{1}{2},y+1,z}+{\bm S}_{x+\frac{1}{2},y+1,z}\right) 
\label{standard_hamiltonian}\end{aligned}$$ and $$\begin{aligned}
  \label{eq:118}
  H_2 & = &  \sum_{xyz}\Big\{ J'' {\bm S}_{x,y,z}\cdot {\bm S}_{x,y,z+1} \\
& & + D (-1)^z {\hat z} \cdot {\bm S}_{x,y,z}\times \big(
  {\bm S}_{x-\frac{1}{2},y+1,z}-{\bm S}_{x+\frac{1}{2},y+1,z}\big) \Big\}.
\nonumber \end{aligned}$$

Review of low energy properties of Heisenberg chains {#sec:review-low-energy}
----------------------------------------------------

We give a brief synopsis of known results on the low energy theory of the single Heisenberg chain in a field, $$\begin{aligned}
  \label{eq:singlechain}
  H_{1d} & = & J \sum_x {\bm S}(x)\cdot {\bm S}(x+1) - h \sum_x S^z(x).\end{aligned}$$ Here $x$ are taken as integers, and we have taken the $z$ axis in spin space along the field. In Eq. (\[eq:singlechain\]) the magnetization $M = \sum_x \frac{1}{N}S^z(x) = \langle S^z(x) \rangle$ is conserved ($N$ is the number of spins), and so it is convenient to work at fixed $M$. The equilibrium relation between magnetization and field $M(h)$ is known from the Bethe ansatz solution, see Ref. .

For any $M$ less than full saturation, $|M|<1/2$, the low energy theory can be described in Abelian bosonization by a single massless free scalar field $\theta$ and its “dual” $\phi$ (related to the canonical momentum conjugate to 