Negativity for two blocks in the one dimensional Spin 1 AKLT model

TL;DR

This paper analyzes the entanglement, measured by negativity, between two blocks in the 1D spin-1 AKLT model under different boundary conditions, finding that negativity vanishes for separated blocks in most cases.

Contribution

It provides explicit calculations of the reduced density matrices and negativity for various boundary conditions and block configurations in the 1D spin-1 AKLT model, revealing conditions under which entanglement vanishes or persists.

Findings

Negativity vanishes for separated blocks with L ≥ 1 in the non-degenerate ground state case.

In degenerate ground state scenarios, negativity can approach 1/2 depending on eigenstate construction.

Negativity is zero for separated blocks in the periodic boundary condition case.

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 AKLT model. We took the model with two different boundary conditions. We consider the case of $N$ spins 1 in the bulk and one spin 1/2 at each boundary which constitute an unique ground state, and the case of just spins 1, even at the end of the chain, where the degeneracy of the ground state is four. 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 $L\geq 1$ 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 non-complementary 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.