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