Impurity entanglement in the J-J2-δ 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 their similarities and the 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 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 crossover 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 d is introduced. In this case we observe a very rapid reduction in the impurity entanglement as delta is increased.