Boundary and impurity effects on the entanglement of Heisenberg chains

We study entanglement of a pair of qubits and the bipartite entanglement between the pair and the rest within open-ended Heisenberg $XXX$ and $XY$ models. The open boundary condition leads to strong oscillations of entanglements with a two-site period, and the two kinds of entanglements are $180°$ out of phase with each other. The mean pairwise entanglement and ground-state energy per site in the $XXX$ model are found to be proportional to each other. We study the effects of a single bulk impurity on entanglement, and find that there exist threshold values of the relative coupling strength between the impurity and its nearest neighbors, after which the impurity becomes pairwise entangled with its nearest neighbors.

Figure 1

Ground-state nearest-neighbor concurrence (square line) and linear entropy (star line) versus site number for $L=12$ in the Heisenberg $XXX$ model.

Figure 2

Mean concurrence (square line) and mean linear entropy (star line) versus $L$ and $e_0$ in the Heisenberg $XXX$ model.

Figure 3

Ground-state nearest-neighbor concurrence (square line) and linear entropy (circle line) entanglement versus site number for $L=100$ in the Heisenberg $XY$ model.

Figure 4

Mean concurrence (square line) and mean linear entropy (star line) versus $L$ and $e_0$ in the Heisenberg $XY$ model.

Figure 5

Ground-state nearest-neighbor concurrence versus site number for $L=100$ in the Heisenberg $XY$ model with an impurity.

Figure 6

Ground-state nearest-neighbor concurrences $C_{N∕2-1,N∕2}$ (square line), $C_{N∕2,N∕2+1}$ (circle line), $C_{N∕2+1,N∕2+2}$ (diamond line), and $C_{N∕2+2,N∕2+3}$ (star line), as a function of $\alpha$ for the impurity model.