follows $$H=\frac{1}{2}[({\bf s}_1+{\bf S}_1+{\bf s}_2+{\bf S}_2)^2-({\bf
s}_1+{\bf s}_2)^2-({\bf S}_1+{\bf S}_2)^2].$$ Then, by the angular momentum coupling theory, we obtain the ground-state energy per site $e_0=E_0/4=-3/4$. Thus, from Eq. (\[N1212\]), we have ${\cal N}_{12}=1/6$. For $N\ge 5$, it is hard to get analytical results. We next numerically calculate the entanglement for the case of more spins, and also consider the effects of magnetic fields.

Numerical results
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Having obtained analytical results of entanglement in the Heisenberg model with a few spins, we now numerically examine the entanglement behaviors in more general Hamiltonian including more spins and magnetic fields.

### Entanglement versus temperature

We consider the entanglement versus temperature for different number of spins $N$, and the numerical results are plotted in Fig. 2 and Fig. 3. It is clear to see that the ground state exhibits maximal entanglement, and with the increase of temperature, the entanglement monotonically decreases until it reaches zero. The decrease of entanglement is due to the mixture of less entangled excited states when increasing the temperature. The existence of the threshold temperature is also evident. For the case of even number of sites (Fig. 2), the threshold temperature decreases with the increase of $N$. In contrast to 