Engineering antiferromagnetic Heisenberg spin chains for maximizing of the groundstate entanglement

We study the correlation function and concurrence for the eigenstates with zero spin of engineered Heisenberg models to explore the entanglement property. It is shown that the total nearest neighbor (NN) correlation function of zero-spin eigenstates reaches its local extremum when the coupling strength is uniform, and correspondingly the groundstate entanglement of $d$-D cubic AF Heisenberg model is locally maximized. Moreover, numerical calculations for a $N$-site quantum spin ring with cosinusoidally modulated exchange coupling, i.e. $J_i=J(1+\cos(2\pi i/N))$, indicate that the uniform coupling is not the unique optimal distribution for maximizing the groundstate entanglement and this modulation of interactions can induce the longer range entanglement.