visualizations/zs.md

ground state is two-fold degenerate, and when \(B > B_{0}\) the ground state becomes non-degenerate and the corre- spending wave function is given by \(\left|\psi _ {0}\right| =\left[1-\frac{1}{2}, 1\right]\). (39) which is obviously of no entanglement. Then, the ground- state negativity forms a plateau when \(B<B_{0}\). The jump of negativity at \(B=B_{0}\) is due to the level crossing. For \(N=2\), the eects of magnetic fields on entanglement are studied by level crossing. For instance, for \(N=4\), there are two level crossing, and the entanglement displays two jumps. III. EFFECTS OF NEXT-NEAREST-NEIGHBOR INTERACTIONS ON ENTANGLEMENT We have studied the eects of einite temperature and magnetic fields on entanglement, and now consider the model containing two kinds of spins, spin 4 and 1, alter- nating on a ring with antiferromagnetic exchange cou- pling between both the NN spins and the NN Hamiltonians. The Hamiltonian can be expressed as \(H=J_{1} \sum _ {i=1}^{N}(\mathbf {s } _ {i} \cdot \mathbf {s } _ {i+1}+\mathbf {s } _ {i+1} \cdot \mathbf {s } _ {i+1})+\frac {N^{2}}{2}J_{2} \sum _ {i=1}^{N}(\mathbf {s } _ {i+1} \cdot \mathbf {s } _ {i+1}+\mathbf {s } _ {i+1} \cdot \mathbf {s } _ {i+1})\), (40) where the \(\mathbf {s } _ {i}\) and \(\mathbf {s } _ {i}\) are spin 1/2 and spin 1 operators in the rth cell. The Hamiltonian characterizes the NN exchange coupling and the