$\theta$) $$\label{eq:duals}
[\theta(x), \phi(x')] = - i \Theta(x-x'),$$ with $\Theta(x)$ the Heaviside step function. The low energy Hamiltonian is then $$\label{eq:H-bos-0}
  H_0 = \int dx ~\frac{v}{2} \Big((\partial_x \phi)^2 + (\partial_x
  \theta)^2\Big)   ,$$ where the velocity $v$ depends on $M$, see Fig. 9 of Ref. . At a given $M$, the fluctuations of the “longitudinal” spin component along the field axis are gapless at wavevectors $k_x=0,\pi\pm
2\delta$, with $\delta=\pi M$. Similarly, the “transverse” spin fluctuations perpendicular to the field axis are gapless at $k_x=\pm 2\delta, \pi$. The lattice spin operator is decomposed thereby according to $$\begin{aligned}
  \label{eq:1dspindecomp}
  S^z(x) & \sim & M + \mathcal{S}^z_0(x) \\ &&+ e^{i(\pi-2\delta)x}
    \mathcal{S}^z_{\pi-2\delta}(x) 
     +e^{-i(\pi-2\delta)x}
    \mathcal{S}^z_{\pi+2\delta}(x) , \nonumber \\
  S^+(x) & \sim &  e^{-i2\delta x} \mathcal{S}^+_{2\delta}(x)
    + e^{i 2\delta x} \mathcal{S}^+_{-2\delta}(x) + (-1)^x
  \mathcal{S}^+_\pi(x). \nonumber\end{aligned}$$ Here the scaling operators $\mathcal{S}^z_0, \mathcal{S}^z_{\pi\pm
  2\delta}$ describe longitudinal spin fluctuations, and $\mathcal{S}^\pm_{\pm 2\delta}, \mathcal{S}^\pm_\pi$ transverse ones. These should be assumed to vary slowly with $x$ (and time). Note that the operators $\mathcal{S}^\mu_k$ ($\mu=z, \pm$) do not mean the Fourier components of $S^\mu(x)$. They can be expressed in terms of bosonic fields as follows $$\begin{aligned}
  \label{eq:bosonizespinops}
  \mathcal{S}^z_0(x) & = & \beta^{-1}\partial_x \phi, \\
  \mathcal{S}^z_{\pi-2\delta}(x) & = & -\frac{i}{2} A_1 e^{-2\pi i \phi/\beta},
  