with $m_x^A=m_x^B=0$ and $m_z^A=m_z^B$, implying the staggered magnetization $M_y=\langle \frac12 (S_y^A-S_y^B)\rangle$. These magnetizations are shown in Fig. \[figXYX\](b) for the two different phases for truncation dimension $D=6$. There is a clear jump discontinuity at $\Delta_y=1$ for each of the staggered magnetizations $M_x$ and $M_y$. This behavior persists with increasing truncation dimension $D$, indicating a discontinuous phase transition at $\Delta_y=1$.

The GE per site is shown for the same parameter range in Fig. \[figXYX\](c) with increasing truncation dimension. The characteristic cusp occurs as $\Delta_{y}$ varies across the critical point $\Delta_{y}=1$. In contrast to the local order parameter, which is discontinuous, the GE is continuous. In this case the GE thus does not detect the discontinuous behavior at the phase transition point.

2D spin-$1$ XXZ model
---------------------

The spin-$1$ XXZ model is defined on the square lattice by the hamiltonian $$H =
  \sum_{( \vec r, \vec r')} \left( S^{[\vec r]}_x S^{[\vec r']}_x
    +  S^{[\vec r]}_y S^{[\vec r']}_y
    + \Delta S^{[\vec r]}_z S^{[\vec r']}_z \right),$$ where now $S^{[\vec r]}_{\alpha} (\alpha=x,y,z)$ are spin-$1$ operators at site $\vec r$, with again the summation running over all nearest neighbor pairs on the square lattice. For anisotropic exchange interaction parameter $\Delta=1$ the hamiltonian has $SU(2)$ symmetry and 