visualizations/gt.md

ground state is two-fold degenerate, and when (B>B_{\text{th}}) the ground state becomes non-degenerate and the corresponding wave function is given by

[|\psi_{0}\rangle=|-\frac{1}{2},-1\rangle,] (39)

which is obviously of no entanglement. Then, the ground-state negativity forms a platform when (0<B<B_{\text{th}}). The jump of negativity at (B=B_{\text{th}}) is due to the level crossing. For (N>2), the effects of magnetic fields on entanglement can be also explained 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 effects of finite temperature and magnetic fields on entanglement, and now consider the model containing two kinds of spins, spin (\frac{1}{2}) and (1), alternating on a ring with antiferromagnetic exchange coupling between both the NN spins and the NNN spins. The Hamiltonian can be expressed as

[H=J_{1}\sum^{N/2}{i=1}\big{(}{\bf s}{i}\cdot{\bf S}{i}+{\bf S}{i}\cdot{\bf s }{i+1}\big{)}+J{2}\sum^{N/2}{i=1}\big{(}{\bf s}{i}\cdot{\bf s}{i+1}+{\bf S }{i}\cdot{\bf S}_{i+1}\big{)},] (40)

where the ({\bf s}{i}) and ({\bf S}{i}) are spin-1/2 and spin-1 operators in the (i)th cell. (J_{1}) characterizes the NN exchange coupling and (J_{2}) the NNN coupling. We consider the antiferromagnetic interaction by taking (J_{1},J_{2}>0). (N) is the total number of spins and here we