Impurity Entanglement in the $J-J_2-\delta$ Quantum Spin Chain

The contribution to the entanglement of an impurity attached to one end of a $J-J_2-delta$ quantum spin chain (S=1/2) is studied. Two different measures of the impurity contribution to the entanglement have been proposed: the impurity-entanglement-entropy S_{imp} and the negativity N. The first, S_{imp}, is based on a subtractive procedure where the entanglement-entropy in the absence of the impurity is subtracted from results with the impurity present. The other, N, is the negativity of a part of the system separated from the impurity and the impurity itself. In this paper we compare the two measures and discuss similarities and differences between them. In the $J-J_2-\delta$ model it is possible to perform very precise variational calculations close to the Majumdar-Ghosh-point (J_2 = J / 2 and \delta = 0) where the system is gapped with a two-fold degenerate dimerized ground-state. We describe in detail how such calculations are done and how they can be used to calculate N as well as S_{imp} for any impurity-coupling J_K. We then study the complete cross-over in the impurity entanglement as J_K is varied between 0 and 1 close to the Majumdar-Ghosh-point. In particular we study the impurity entanglement when a staggered nearest-neighbour-interaction proportional to $\delta$ is introduced. In this case, the two-fold degeneracy of the ground-state is lifted leading to a very rapid reduction in the impurity entanglement as $\delta$ is increased.