Negativity for two blocks in the one-dimensional spin-1 Affleck-Kennedy-Lieb-Tasaki model

Abstract

In this paper we compute the entanglement, as quantified by negativity, between two blocks of length ${L}_{A}$ and ${L}_{B}$, separated by $L$ sites in the one-dimensional spin-1 Affleck-Kennedy-Lieb-Tasaki (AKLT) model. We took the model with two different boundary conditions. We consider the case of $N$ spin-1 particles in the bulk and one spin-1/2 particle at each boundary, which constitute a unique ground state, and the case of just spin 1, even at the end of the chain, where the degeneracy of the ground state is 4. In both scenarios we made a partition consisting of two blocks $A$ and $B$, containing ${L}_{A}$ and ${L}_{B}$ sites, respectively. The separation of these two blocks is $L$. In both cases we explicitly obtain the reduced density matrix of the blocks $A$ and $B$. We prove that the negativity in the first case vanishes identically for $L\ensuremath{\geqslant}1$, while in the second scenario it may approach a constant value $N=1/2$ for each degenerate eigenstate depending on the way one constructs these eigenstates. However, as there is some freedom in constructing these eigenstates, vanishing entanglement is also possible in the latter case. Additionally, we also compute the entanglement between noncomplementary blocks in the case of periodic boundary conditions for the spin-1 AKLT model for which there is a unique ground state. Even in this case, we find that the negativity of separated blocks of spins is zero.