value of $J_2$.

From another point of view, we can read the threshold temperature $T_\text {th}$ versus different $J_2$ from the curve in the $J_2-T$ plane. When $J_2$ increases, $T_\text {th}$ decreases, and when $J_2$ crosses about $0.3758$, ${\cal N}_{1/2,1}$ will disappear at any temperature.

Next, we consider the entanglement between NNN spins. In Fig. 7, we plot the negativity ${\cal N}_{1/2,1/2}$ as a function of the temperature and $J_2$. We can see that, before $J_2$ reaches the value about $J_2=0.5$, ${\cal N}_{1/2,1/2}$ keeps being zero at any temperature. And in the region $J_2>0.5$, the ${\cal
N}_{1/2,1/2}$ can be enhanced by the increasing NNN interaction. This is a result from the competition of two kinds of exchange interactions. The thermal fluctuation all along suppresses the entanglement. So, from the curve lying on the $J_2-T$ plane which corresponds to the boundary of the nonzero and zero values of ${\cal N}_{1/2,1/2}$, we may find that the higher the temperature is, the larger the threshold $J_\text {2th}$ will be. From another point of view, the $T_\text {th}$ increases as $J_2$ increases.

![Negativity ${\cal N}_{1/2,1/2}$ versus both the exchange interaction coefficient $J_2$ and the temperature in four-spin system.](next_fig5.eps){width="45.00000%"}

In Fig. 8, we plot the 