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arXiv:1011.3277 · Physical Review B

Topological disentangler for the valence-bond-solid chain

Kouichi Okunishi

Published Mar 21, 2011• 16 citations
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Abstract

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.

DOI
10.1103/physrevb.83.104411

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We discuss the topological disentangler for S=1S=1S=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=1S=1S=1 Heisenberg chain.
Two-dimensional (spin-2) Affleck-Kennedy-Lieb-Tasaki (AKLT) type valence bond solids on a square lattice are known to be symmetry-protected topological (SPT) gapped spin liquids [S. Takayoshi, P. Pujol, and A. Tanaka Phys. Rev. B 94, 235159 (2016)]. Using the projected entangled pair state framework, we extend the construction of the AKLT state to the case of SU(3)\text{SU}(3)SU(3), relevant for cold atom systems. The entanglement spectrum is shown to be described by an alternating SU(3)\text{SU}(3)SU(3) chain of ``quarks'' and ``antiquarks'', subject to exponentially decaying (with distance) Heisenberg interactions, in close similarity with its SU(2)\text{SU}(2)SU(2) analog. We discuss the SPT feature of the state.
Disentangling transformations play a central role in the classical simulation of quantum many-body systems, yet their analytic structure and underlying mechanism remain largely unexplored. Here, we study the structure of the disentangler in the Haldane phase of spin-1 systems using generalized Clifford circuits. To this end, we extend the Clifford-circuit-augmented matrix product states (CAMPS)-based density-matrix renormalization group (DMRG) method to spin-1 systems. Within this framework, we find that the local disentanglers optimized for the Haldane phase implement the generalized Kramers--Wannier (KW) transformation, and we analytically verify its optimality for the Affleck--Kennedy--Lieb--Tasaki (AKLT) state. Beyond reducing entanglement, the KW transformation maps the Haldane phase to a phase with spontaneously broken Z2\mathbb{Z}_{2}Z2​ symmetry. This mapping is distinct from the Kennedy--Tasaki transformation and provides a new unitary route from symmetry-protected topological order to symmetry breaking.
Symmetry-protected topological phases are defined and classified in reference to symmetries that are described by groups. The author argues that there exist such phases that are not captured by this classification. To support this claim a qqq-deformation of the Affleck-Kennedy-Lieb-Tasaki model is discussed, which exhibits quantum group and duality-type symmetries but breaks all standard group symmetries known to protect the Haldane phase. Its nontrivial topology is linked to a twofold degeneracy in a suitably deformed notion of entanglement spectrum.
We investigate the entanglement properties of the valence-bond-solid (VBS) state defined on two-dimensional lattices, which is the exact ground state of the Affleck-Kennedy-Lieb-Tasaki model. It is shown that the entanglement entropy obeys an area law and the nonuniversal prefactor of the leading term is strictly less than ln2\mathrm{ln}2ln2. The analysis of entanglement spectra for various lattices reveals that the reduced density matrix associated with the VBS state is closely related to a thermal density matrix of a holographic spin chain, the spectrum of which is reminiscent of that of the spin-1/2 Heisenberg chain. This correspondence is further supported by comparing the entanglement entropy in the holographic spin chain with conformal field theory predictions.
Applying a symmetric bulk bipartition to the one-dimensional Affleck-Kennedy-Lieb-Tasaki valence-bond solid (VBS) states for the integer spin-SSS Haldane gapped phase, we can create an array of fractionalized spin-SSS/2 edge states with the super unit cell lll in the reduced bulk system, and the topological properties encoded in the VBS wave functions can be revealed. The entanglement Hamiltonian (EH) with even lll corresponds to the quantum antiferromagnetic Heisenberg spin-SSS/2 model. For the even integer spins, the EH still describes the Haldane gapped phase. For the odd integer spins, however, the EH just corresponds to the quantum antiferromagnetic Heisenberg half-odd integer-spin model with spinon excitations, characterizing the critical point separating the topological Haldane phase from the trivial gapped phase. Our results thus demonstrate that the topological bulk property not only determines its fractionalized edge states but also the quantum criticality associated with the topological phase, where the elementary excitations are precisely those fractionalized edge degrees of freedom confined in the bulk of the topological phase.
Abstract The Haldane phase is the prototype of symmetry protected topological (SPT) phases of spin chain systems. It can be protected by several symmetries having in common the degeneracy of the entanglement spectrum (ES). Here we explore in depth this degeneracy for the spin-1 Affleck–Kennedy–Lieb–Tasaki and bilinear-biquadratic Hamiltonians and show the emergence of a bulk-edge correspondence that relates the low energy properties of the entanglement Hamiltonian of a periodic chain and that of the physical Hamiltonian of an open chain. We find that the ES can be described in terms of two spins-1/2 behaving as the effective spins at the end of the open chain. In the case of non-contiguous partitions, we find that the entanglement Hamiltonian is given by the spin-1/2 Heisenberg Hamiltonian, which suggests a relationship between SPT phases and conformal field theory. We finally investigate the string order parameter and the relation with the bulk-edge correspondence.