the unitary element of the broken $Z_2$ symmetry group. Thus two degenerate ground states are detected and distinguished by the sign of the local order parameter $\langle S_x\rangle$, with the amplitude associated with each of the two degenerate ground states having the same value. The order parameter $\langle S_x\rangle$ is shown in Fig. \[fig1\](a). Phase transition points are identified at the values shown in Table I. Near these points the amplitude of $\langle S_x\rangle$ is observed to continuously approach zero, indicative of a continuous phase transition.

On the other hand, the estimates for the GE per lattice site for the ground states are shown in Fig. \[fig1\](b). For each value of $D$ the GE curve has a maximal value at $\lambda_c(D)$, which indicate the phase transition points. These values, shown in Table I, agree with the values detected with the local order parameter. Our results are consistent with the previous studies [@hl; @gehlw; @thesis]. The GE is seen to be continuous near the maximal points. The cusp-like behavior at the critical point is a characteristic feature of the GE at a continuous phase transition. The two degenerate ground states for $\lambda < \lambda_c$ can also be distinguished via the initial 