this behavior, for the case of odd number of spins, as seen from Fig. 3, the threshold temperature increases with the increase of $N$.

![ Negativity versus temperature for different even number of spins in the Heisenberg model.](mixfig2.eps){width="45.00000%"}

![ Negativity versus temperature for different odd number of spins in the Heisenberg model.](mixfig3.eps){width="45.00000%"}

### Effects of magnetic fields

We now examine the effect of magnetic fields on entanglement. The Heisenberg Hamiltonian with a magnetic field along $z$ direction is given by $$H_1=H_0+B\sum_{i=1}^{N/2} \big({s}_{iz}+{S}_{i,z}\big), (N\in
\text{even}).$$ For $N=2$, the analytical result of negativity can be obtained (see Appendix A).

![Negativity versus the magnetic field for $T=0.05$.](mixfig4.eps){width="45.00000%"}

In Fig. 4, we plot the negativity versus the magnetic field at a low temperature. For the two-spin case, with the increase of the magnetic field, the negativity rapidly reach a platform, and then after a certain magnetic field $B_\text{th}$, it jumps down to zero, indicating no entanglement. When $B=0$, the ground state is two-fold degenerate with ${\cal N}=1/3$. For a small $B>0$, the system is no longer degenerate, and the ground state is given by $$|\psi_0\rangle=\frac{\sqrt{6}}{3}|\frac{1}{2},-1\rangle-\frac{\sqrt{3}}{3}|-\frac{1}{2},0\rangle,$$ which is of the Schmidt form. For a state written as its Schmidt form $$|\psi\rangle=\sum_n c_n|e_n\rangle|f_n\rangle,$$ the negativity is 