{-\frac{10}{3}e^{-\frac{5}{4}\beta}-e^{\frac{3}{4}\beta}+\frac{1}{3}e^{\frac{7}{4}\beta}}\Big]/Z.$$ It is evident that the negativity becomes 1/3 in the limit of $T\rightarrow 0$. From the expression of the negativity, the threshold value of temperature after which the entanglement vanishes can be estimated as $$T_{\text{th}}\approx 1/\ln3.2719\approx0.7609.$$

To examine the entanglement between two spin halves, we trace out the spin 1 system and get the reduced density matrix $\rho_{13}$ as follows $$\rho_{13}=\left(
\begin{array}{llll}
a_1& 0&  0&   0\\
0&   a_2&b&   0\\
0&   b&  a_2& 0\\
0&   0&  0&   a_1
\end{array} \right),\label{rho13}$$ with the matrix elements given by $$\begin{aligned}
a_1&=&\left(\frac{5}{3}e^{-\frac{5}{4}\beta}+e^{\frac{3}{4}\beta}+\frac{1}{3}e^{\frac{7}{4}\beta}\right)/Z\nonumber\\
a_2&=&\left(\frac{5}{6}e^{-\frac{5}{4}\beta}+2e^{\frac{3}{4}\beta}+\frac{1}{6}e^{\frac{7}{4}\beta}\right)/Z\nonumber\\
b&=&\left(\frac{5}{6}e^{-\frac{5}{4}\beta}-e^{\frac{3}{4}\beta}+\frac{1}{6}e^{\frac{7}{4}\beta}\right)/Z.
\label{ele}\end{aligned}$$ After taking the partial transpose, we can get $\rho^T_{13}$ $$\rho^T_{13}=\left(
\begin{array}{llll}
a_1& 0&  0&   b\\
0&   a_2&0&   0\\
0&   0&  a_2& 0\\
b&   0&  0&   a_1
\end{array} \right).\label{rhoT13}$$ Then, the negativity is readily obtained as $${\cal N}(\rho_{13})=\max[0,|b|-a_1].$$ It is straightforward to check that the negativity is always zero. Or, from another way, all the eigenvalues of the matrix $\rho^T_{13}$ are obtained as $$\begin{aligned}
\lambda_1&=&\left(\frac{5}{2}e^{-\frac{5}{4}\beta}+\frac{1}{2}e^{\frac{7}{4}\beta}\right)/Z,\nonumber\\
\lambda_2&=&\left(\frac{5}{6}e^{-\frac{5}{4}\beta}+2e^{\frac{3}{4}\beta}+\frac{1}{6}e^{\frac{7}{4}\beta}\right)/Z,\nonumber\\
\lambda_3&=&\lambda_4=a_2.\end{aligned}$$ Obviously the negativity vanishes here, in other words there is no entanglement between the two spin halves.

The ground-state negativity ${\cal N}_{12}={\cal
N}(\rho_{12})$=1/3 and ${\cal N}_{13}={\cal N}(\rho_{13})$=0, here ${\cal N}_{12}$ denotes the negativity between the $1_\text{th}$ and $2_\text{th}$ spin on the chain and 