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

TL;DR

This paper investigates impurity-induced entanglement in a $J-J_2- ext{delta}$ quantum spin chain, comparing two measures of entanglement and analyzing how impurity coupling and staggered interactions affect entanglement properties.

Contribution

It introduces and compares two measures of impurity entanglement, providing detailed variational calculations near the Majumdar-Ghosh point and analyzing effects of staggered interactions.

Findings

Impurity entanglement varies with impurity coupling J_K.

Staggered interactions rapidly reduce impurity entanglement.

Precise variational calculations are feasible near the Majumdar-Ghosh point.

Abstract

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.