Hamiltonian of the fully anisotropic XYZ spin-$\frac{1}{2}$ chain, $$\label{ham-XXZ}
\hat{H}_{XYZ}= \sum_{r} \Big\{J_x S^{x}_{r}S^{x}_{r+1}
+J_yS^{y}_{r}S^{y}_{r+1} + J_z  S^{z}_{r}S^{z}_{r+1}\Big\}.$$ We will assume $|J_z|<J_y\le J_x$. Denoting $\Delta=J_z/J$, $J=(J_x+J_y)/2$, and $\gamma= (J_x-J_y)/(2J)$, we rewrite the Hamiltonian of the XYZ chain as follows, $$\hat{H}_{XYZ}= J \sum_{r} \Big\{ (1+\gamma)S^{x}_{r}S^{x}_{r+1}
+ (1-\gamma)S^{y}_{r}S^{y}_{r+1} + \Delta  S^{z}_{r}S^{z}_{r+1}\Big\}.$$

We will put $\hbar=1$ and measure energy in units of $J$. Spin-wave velocity for the gapless, $\gamma=0$, case is $$\label{velo}
v=\frac{Jr_0 \sin{(\pi \eta)}}{2(1-\eta)},$$ where $r_0$ is the lattice constant and $\eta=1-({\arccos{\Delta}})/{\pi}$. It is convenient to fix the spin-wave velocity equal to unity (hence also make dimensionless) for $\gamma=0$, independently of $\Delta$. For this, we will fix $$\label{fixation}
Jr_0=2\frac{1-\eta}{\sin{(\pi\eta)}}.$$

In the following we will use the exact solution of the XYZ chain [@Baxter82; @JohnsonKrinskyMcCoy73]. In particular we will be interested in the limit $\gamma\to 0$, and take the so called scaling limit of the XYZ chain, where the spin gap behaves as [@Luther76] $$\begin{aligned}
\label{Gap}
M_{XYZ}&=&2 J_x \frac{\sin({\pi \eta})}{ 1-\eta} \left( \frac{\sqrt{J_x^2-J_y^2}  }{4\sqrt{J_x^2-J^2_z}} \right)^{\frac{1}{1-\eta}} \nonumber\\
&\simeq& 2 J \frac{\sin({\pi \eta})}{ 1-\eta} \left( \frac{\gamma}{4(1-\Delta^2)} \right)^{\frac{1}{2(1-\eta)}} \nonumber\\
&=&\frac{4}{r_0} \left( \frac{\gamma}{4\sin^2(\pi \eta)} \right)^{\frac{1}{2(1-\eta)}}\nonumber\\
&=&4\left( \frac{\gamma_r}{4\sin^2(\pi \eta)} \right)^{\frac{1}{2(1-\eta)}},\end{aligned}$$ where $\gamma_r= \gamma r_0^{-2(1-\eta)}$ and to arrive from the second to the third line we used Eq. (\[fixation\]). The 