negativity ${\cal N}_{1,1}$ versus $T$ and $J_2$. In the region of $J_2<0.25$, ${\cal N}_{1,1}$ is zero at any temperature. When $J_2>0.25$, the increasing NNN exchange interaction $J_2$ enhances the negativity and exhibits two particular flat roofs. With the temperature rises, ${\cal
N}_{1,1}$ is suppressed to zero. Also we can consider the threshold $T_\text {th}$ and $J_\text {2th}$ from the critical curve on the $J_2-T$ plane, and the $J_\text {2th}$ also behaves as an increasing function of the temperature. We can see the nonzero region of ${\cal N}_{1,1}$ is much larger than ${\cal
N}_{1/2,1/2}$. But here it should be pointed out that, because ${\cal N}_{1,1}>0$ only gives a sufficient condition for entangled state, we can not definitely say that the state in the area of zero negativity is not entangled.

![Negativity ${\cal N}_{1,1}$ versus both the exchange interaction coefficient $J_2$ and the temperature in four-spin system.](next_fig6.eps){width="45.00000%"}

Numerical results of negativity for more spins
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In this section, we present numerical results of negativity for more spins, and first consider the low-temperature case.

### Low-temperature case

![Negativity versus the exchange interaction $J_2$ at the temperature $T=0.02$ in six-spin system. The ground and the first two excited energy levels versus the $J_2$ are 