2\delta}$       $1/4\pi R^2$       $1/2$          $1$
    $\mathcal{S}^\pm_{\pm 2\delta}$    $\pi R^2+1/4\pi R^2$    $1$          $5/4$
        $\mathcal{S}^\pm_{\pi}$             $\pi R^2$         $1/2$         $1/4$

  : Scaling dimensions of scaling fields associated with spin fluctuations in the one dimensional Heisenberg chain at magnetization $M$. The third and fourth columns give the scaling dimensions in the limit of zero and full polarization, respectively.[]{data-label="tab:dims"}

In addition to the scaling fields above, which appear in the expansion of spin operators, we will also make use of the spin current, for the component of spin along the field axis. This has the form $$\begin{aligned}
  \label{eq:19}
  J^z(x) & = & \frac{-i}{2} \left[ S^+(x) S^-(x+1) -
    S^-(x)S^+(x+1)\right]\nonumber \\
&&  \sim \mathcal{J}^z(x).\end{aligned}$$ Like the spin density $\mathcal{S}_0^z$, this has a simple bosonization formula, $$\label{eq:20}
  \mathcal{J}^z = \frac{v}{\beta J} \partial_x \theta.$$ We note that at $M=0$, $v/J=\pi/2$ takes a simple value, but at $M>0$ the coefficient decreases continuously.

It may also be useful to connect with the limit of zero field $M=\delta=0$, which is probably more familiar. Here the Hamiltonian has SU(2) symmetry. In this limit, the three operators ${\rm
  Re}[\mathcal{S}_\pi^+]$, ${\rm Im}[\mathcal{S}_\pi^+]$, and ${\rm
  Re}[\mathcal{S}^z_{\pi-2\delta}]$ become unified into the three components of the Néel field ${\bm N}$ (scaling dimension 1/2), while the other three operators, 