In Fig. \[fig3\](a) we show the local magnetizations for sublattices A and B defined by $m_{\alpha}^A=\langle S_{\alpha}^A\rangle$ and $m_{\alpha}^B=\langle S_{\alpha}^B\rangle$ for $\alpha=x,y,z$. For $h<4$, $m_\alpha^A = -m_\alpha^B$ for $\alpha=x,y$ with $m_z^A=m_z^B$. For $h>4$, $m_\alpha^A = m_\alpha^B = 0$ for $\alpha=x,y$ with $m_z^A=m_z^B =\frac12$. The latter is indicative of the fully polarized state.

In Fig. \[fig3\](b) we show the staggered magnetization parameters $M_{xy}=\sqrt{\langle \frac12(S_x^A-S_x^B)\rangle^2+\langle \frac12(S_y^A-S_y^B)\rangle^2}$ and $M_z=\langle \frac12(S_z^A+S_z^B)\rangle$ as functions of the magnetic field. The local order parameter $M_{xy}$ decreases with increasing $h$, with $M_{xy} \to 0$ when $h>4$. On the other hand, $M_z$ increases with increasing $h$, with $M_{z} \to \frac12$ for $h>4$, which corresponds to a continuous phase transition from a Néel phase to the fully polarized phase. As mentioned previously, for $h<4$, infinitely degenerate ground states exist corresponding to the $x$-$y$ plane U$(1)$ symmetry breaking, which is indicated from the random-like magnetization on the sublattices. As for the previous models, the numerical results indicate that the different degenerate ground states for fixed $h$ give the same value for the GE per site, as Fig. \[fig3\](c) shows. In Fig. \[fig3\](c), the GE per site clearly decreases with increasing $h$, indicating decreasing entanglement as $h\to 4$. In fact, zero GE 