considered to be a threshold value, after which the NN entanglement vanishes.

Having studied ground-state entanglement, now we make a short discussion of entanglement of excited states. We consider the first excited state. In the region $J_1/4<J_2<3J_1/8$, the first excited energy is given by $$E_1=-3J_1+\frac{5}{2}J_2,$$ from that we can get $\langle{\bf s}_1\cdot{\bf S}_1\rangle=-3/4$ and $N_{1/2,1}=1/6$. While in the rest region the negativity of the first excited state is zero.

### Thermal entanglement

Next, we consider the thermal entanglement. From Eq. (\[E\]), the partition function can be obtained as $$\begin{aligned}
\label{PF}
Z&=&5e^{-\frac{1}{2}\beta J_2}+6e^{\frac{7}{2}\beta
J_2}+e^{\frac{11}{2}\beta J_2}\nonumber\\
&+&e^{-\frac{5}{2}\beta J_2}(7e^{-2\beta J_1}+5e^{\beta J_1}+3e^{3\beta J_1})\nonumber\\
 &+&e^{\frac{3}{2}\beta J_2}(5e^{-\beta J_1}+3e^{\beta J_1}+e^{2\beta
 J_1}),\end{aligned}$$ The correlator $\langle{\bf s_1}\cdot {\bf S_1} \rangle$ at finite temperature can be computed from the partition function via the following relation $$\label{r1}
\langle{\bf s_1}\cdot {\bf S_1} \rangle=-\frac{1}{4\beta
Z}\frac{\partial Z}{\partial J_1},$$

Substituting (\[PF\]) to Eq. (\[r1\]) yields $$\begin{aligned}
\langle{\bf s_1}\cdot {\bf S_1} \rangle&=&-\frac{1}{4Z}\big [
e^{-\frac{5}{2}\beta J_2}(-14e^{-2\beta J_1 }+5e^{\beta J_1}+9e^{3\beta J_1})\nonumber\\
&+&e^{\frac{3}{2}\beta J_2}(-5e^{-\beta J_1}+3e^{\beta
 J_1}+2e^{2\beta J_1})\big ].\end{aligned}$$ After substituting the above equation into (\[cor\]), we may get analytical expression of the negativity ${\cal N}_{1/2,1}$ at finite temperatures. The negativity is a function of $J_1$, $J_2$ and $T$.

*Low-temperature case*:

![Negativity versus the exchange interaction $J_2$ at the temperature 