Topological disentangler for the valence-bond-solid chain

We discuss the topological disentangler for $S=1$ quantum spin chains in the Haldane phase. We first point out that the Kennedy-Tasaki (KT) nonlocal unitary transformation is the perfect disentangler for the Affleck-Kennedy-Lieb-Tasaki model. We then demonstrate that the KT transformation can be reconstructed as an assembly of pair disentanglers. Finally, we show that the KT transformation can be regarded as a topological disentangler, which selectively disentangles the twofold degeneracy in the entanglement spectrum of the $S=1$ Heisenberg chain.

Figure 1

Schematic diagram of the KT transformation for a 4 spin system. The KT transformation is represented as an assembly of the disentanglers for all pairs.

Figure 2

Disentangling the spin at $n$th site.

Figure 3

Entanglement spectrum of the half-infinite reduced density matrix for the $S=1$ Heisenberg chain (open circles) and the Kennedy-Tasaki transformed chain (solid circles). The spectrum is normalized by the largest eigenvalue.