visualizations/gt2.md

e 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.

In Fig. 8, we plot the 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 s