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