From Eq. (\[H4\]), it is obvious that in addition to the the SU$(2)$ symmetry, there also exists an exchange symmetry, namely, exchanging two spin halves or two spin ones yields invariant Hamiltonian. Then, the correlator $\langle {\bf s}_i\cdot{\bf
S}_i \rangle$ between any NN spins are the same. Therefore, we can get the correlator $\langle {\bf s}_i\cdot{\bf S}_i \rangle$ from the ground-state energy via the Hellmann-Feynman theorem.

When $0\leq J_2<J_1/4$, after applying the Hellmann-Feynman theorem to the ground-state energy, we obtain $$\langle{\bf s}_1\cdot{\bf S}_1\rangle=\frac{1}{4}\frac{\partial
E_0} {\partial J_1}=-\frac{3}{4}.$$ Due to the SU$(2)$ symmetry in our system, we have the following relation between the negativity and the correlator $\langle{\bf
s}_1\cdot{\bf S}_1\rangle$[@Schliemann], $${\cal N}_{1/2,1}=\max\big\{0, -\frac{1}3-\frac{2}3\langle{\bf
s}_1\cdot{\bf S}_1\rangle \big \},\label{cor}$$ where we use ${\cal N}_{1/2,1}$ to denote the negativity between the NN spins. Thus we obtain $${\cal N}_{1/2,1}=1/6.$$ In other regions, we find that ${\cal N}_{1/2,1}=0$ for $J_2>J_1/4$.

From the above analytical results, we can find that the negativity between the NN spins is not a continuous function of parameter $J_2$. It jumps from the value $\frac{1}{6}$ down to zero at $J_2=0.25$. Hence, we can see that the NNN interaction may deteriorate the entanglement of the NN spins. The critical point of $J_2=0.25$ may be 