the ground state is infinitely degenerate. For this model there is a quantum phase transition at $\Delta=1$ [@spin1]. This phase transition is clearly marked in Fig. \[figXXZ\_1\], which shows plots of the single-copy entanglement for the spin-$1$ XXZ model.

We also calculate the components of the local magnetization $m$ for sublattices $A$ and $B$: $m_{\alpha}^A=\langle S_{\alpha}^A\rangle$ and $m_{\alpha}^B=\langle S_{\alpha}^B\rangle$ for $\alpha=x,y,z$. These are shown in Fig. \[figXXZ\_2\](a) as a function of $\Delta$. For $\Delta<1$, the values of $m_{x}$ and $m_{y}$ have opposite signs on each sublattice, with $m_z^A=m_z^B$. For $\Delta>1$, $m_{x}=m_{y}=0$ on each sublattice, with $m_z^A=-m_z^B$. This indicates a local order with staggered magnetization in the $z$ phase characterized by $
M_{z}=\langle \frac12 (S_z^A-S_z^B)\rangle.
$ For different values of magnetic field strength $h$, the magnetization for the $x$ and $y$ directions does not change continuously for $\Delta<1$, while the staggered magnetization $
M_{xy}=\sqrt{\langle \frac12(S_x^A-S_x^B)\rangle^2+\langle \frac12(S_y^A-S_y^B)\rangle^2},
$ continuously changes with varying $\Delta$ (see Fig. \[figXXZ\_2\](b)). This indicates the existence of local order defined with the staggered magnetization in the $x$-$y$ plane.

Hence, for $\Delta<1$, the phase is characterized by the antiferromagnetic order parameter $M_{xy}$ in the $x$-$y$ easy plane. Infinitely degenerate ground states exist corresponding to the $x$-$y$ plane $U(1)$ symmetry 