Impurity Entanglement in the J − J2 − δ Quantum Spin Chain

Abstract. The contribution to the entanglement of an impurity attached to one end of a J − J2 − δ 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 Simp and the negativity N. The first, Simp, 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 − J2 − δ model it is possible to perform very precise variational calculations close to the Majumdar-Ghosh-point (J2 = J/2 and δ = 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 Simp for any impurity coupling JK. We then study the complete cross-over in the impurity-entanglement as JK is varied between 0 and 1 close to the Majumdar-Ghosh-point. In particular we study the impurity-entanglement when a staggered nearest-neighbor-interaction proportional to δ is introduced. In this case, the two-fold degeneracy of the ground-state is broken leading to a very rapid reduction in the impurity entanglement as δ is increased