axis. (b) The arrows indicate the direction of the DM vectors along $b$ or $c$ axis. We use the convention (see Eq. ) in which the first spin in the cross product ${\bm D}_{ij}\cdot {\bm S}_i \times
  {\bm S}_j$ is the one with smaller $x$ for the on-chain bonds (i.e. it is on the left side of the bond), and it is the one with smaller $y$ on the diagonal bonds (i.e. it is the lower of the two spins). Note that a different convention is used in Refs. .[]{data-label="fig: DM_distribution"}](DM_distribution.pdf){width="0.8\columnwidth"}

Competing Interactions, Scaling, and Chain Mean Field Theory {#sec:renorm-group-appr}
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We assume that [*all*]{} inter-chain couplings are weak. In this case, a scaling analysis based on a perturbative RG treatment is appropriate. This proceeds in a standard way. One integrates out short-distance modes (i.e. small $x$ or large $k_x$), progressively reducing the large momentum cutoff from its “bare” value $\Lambda$ (of order the inverse lattice spacing, which we take in turn to be $O(1)$) to $\Lambda
e^{-\ell}$, where $\ell \in [0,\infty]$ is the logarithmic RG scaling variable. Equivalently, the corresponding real space cutoff is $L=ae^\ell$, where $a$ is a microscopic length (which we take to be $O(1)$). As we 