Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1109.4202

Natural extension of hidden $Z_2 \times Z_2$ symmetry toward arbitrary integer spin chains

Isao Maruyama

Published Sep 20, 2011• 1 citation
arXivHTMLPDF

Abstract

We show how entangled valence-bond singlet pairs are disentangled partially and totally by the Kennedy-Tasaki transformation which reveals the hidden $Z_2\times Z_2$ symmetry in valence-bond-solid chains as a higher-spin generalization of the previous studies toward the intermediate-$D$ state. The totally disentangled states correspond to four Ising-like states with $Z_2$ variables on the boundary. We present a simple expression of results by using the spin decomposition and the boundary matrix.

DOI
10.48550/arxiv.1109.4202

Related papers

Topological disentangler for the valence-bond-solid chain

Kouichi Okunishi

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.

arXiv:1011.3277•Physical Review B•Mar 21, 2011•16 citations

Hidden symmetry breaking in a generalized valence-bond solid model

Kiyohito Totsuka, Masuo Suzuki

A 'boson' representation for the ground state of the generalized valence-bond solid model is derived. The relation between this expression and the matrix-product representation of Klumper et al. (1991,1992) is discussed. It is shown that the hidden Z 2 *Z 2 -symmetry is partially broken in this model. Elementary excitations of this system have a topological nature similar to that in the Affleck, Kennedy, Lieb and Tasaki (1988) model.

Journal of Physics A Mathematical and General•Oct 7, 1994•20 citations

Entanglement in valence-bond-solid states on symmetric graphs

Hosho Katsura, Naoki Kawashima, Anatol N. Kirillov +2 more

We study quantum entanglement in the ground state of the Affleck–Kennedy–Lieb–Tasaki model defined on two-dimensional graphs with reflection and/or inversion symmetry. The ground state of this spin model is known as the valence-bond-solid state. We investigate the properties of a reduced density matrix of a subsystem which is a mirror image of the other one. Thanks to reflection symmetry, the eigenvalues of the reduced density matrix can be obtained by numerically diagonalizing a real symmetric matrix whose elements are calculated by Monte Carlo integration. We calculate the von Neumann entropy of the reduced density matrix. The obtained results indicate that there is some deviation from the naive expectation that the von Neumann entropy per valence bond on the boundary between the subsystems is ln 2. This deviation is interpreted in terms of the hidden spin chain along the boundary between the subsystems. In some cases where graphs are on ladders, the numerical results are analytically or algebraically confirmed.

arXiv:1003.2007•Journal of Physics A Mathematical and Theoretical•May 27, 2010•25 citations

Hidden Z 2 *Z 2 symmetry in quantum spin chains with arbitrary integer spin

Masaki Oshikawa

The author studies integer S>1 spin chains. He extends the Kennedy-Tasaki nonlocal unitary transformation for S=1 to arbitrary integer S. He shows the main results of Kennedy and Tasaki (1992) are maintained for S>1: Heisenberg-type Hamiltonians are transformed to Hamiltonians of nearest-neighbour interactions with Z 2 *Z 2 symmetry, and the den Nijs-Rommelse string observables are transformed to the ferromagnetic correlation observables. He asserts that in general values of integer S there exist several phases with the hidden Z 2 *Z 2 symmetry breaking. The den Nijs-Rommelse string order parameters, which measure the hidden Z 2 *Z 2 symmetry breaking, are calculated explicitly for several variants of the VBS-type states. In the standard VBS state, the hidden Z 2 *Z 2 symmetry breaks down when S is odd but remains unbroken when S is even. His results for partially dimerized VBS states suggest that the hidden Z 2 *Z 2 symmetry breaking can be used to detect the successive dimerization transitions predicted by Affleck and Haldane (1987). Some new anisotropic VBS-type states are investigated. The result suggests that there are successive phase transitions when he increases the uniaxial anisotropy in a Heisenberg-type model. Other new VBS-type states with long-range order are considered, and their relevance to the phase diagram of the Heisenberg XXZ model and the magnetization process of antiferromagnets is investigated. He introduces an extended string order parameter which possesses a characteristic behavior in the partially dimerized VBS states.

Journal of Physics Condensed Matter•Sep 7, 1992•222 citations

Disentangling topological degeneracy in the entanglement spectrum of one-dimensional symmetry-protected topological phases

Wen-Jia Rao, Guang-Ming Zhang, Kun Yang

One-dimensional valence bond solid (VBS) states represent the simplest symmetry-protected topological phases. We show that their ground state entanglement spectrum contains both topological and nontopological structures. For the $\text{SO}(3)$ symmetric VBS states with odd-integer spins, the twofold topological degeneracy is associated with an underlying ${Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}$ symmetry that protects the corresponding topological phase. In general, for the $\text{SO}(2S+1)$ symmetric VBS states with integer spins $S$, the corresponding protecting symmetry is identified as the ${({Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2})}^{S}$ symmetry, yielding the ${2}^{S}$-fold topological degeneracy. The topological degeneracy and associated protecting symmetry can be identified by a nonlocal unitary transformation, which changes the topological order of the VBS states into conventional ferromagnetic order.

arXiv:1403.0660•Physical Review B•Mar 17, 2014•8 citations

Entanglement spectra of the two-dimensional Affleck-Kennedy-Lieb-Tasaki model: Correspondence between the valence-bond-solid state and conformal field theory

Jie Lou, Shu Tanaka, Hosho Katsura +1 more

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 $\mathrm{ln}2$. 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.

arXiv:1107.3888•Physical Review B•Dec 27, 2011•47 citations
Natural extension of hidden Z2×Z2Z_2 \times Z_2Z2​×Z2​ symmetry toward arbitrary integer spin chains
We show how entangled valence-bond singlet pairs are disentangled partially and totally by the Kennedy-Tasaki transformation which reveals the hidden Z2×Z2Z_2\times Z_2Z2​×Z2​ symmetry in valence-bond-solid chains as a higher-spin generalization of the previous studies toward the intermediate-DDD state. The totally disentangled states correspond to four Ising-like states with Z2Z_2Z2​ variables on the boundary. We present a simple expression of results by using the spin decomposition and the boundary matrix.
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.
We study quantum entanglement in the ground state of the Affleck–Kennedy–Lieb–Tasaki model defined on two-dimensional graphs with reflection and/or inversion symmetry. The ground state of this spin model is known as the valence-bond-solid state. We investigate the properties of a reduced density matrix of a subsystem which is a mirror image of the other one. Thanks to reflection symmetry, the eigenvalues of the reduced density matrix can be obtained by numerically diagonalizing a real symmetric matrix whose elements are calculated by Monte Carlo integration. We calculate the von Neumann entropy of the reduced density matrix. The obtained results indicate that there is some deviation from the naive expectation that the von Neumann entropy per valence bond on the boundary between the subsystems is ln 2. This deviation is interpreted in terms of the hidden spin chain along the boundary between the subsystems. In some cases where graphs are on ladders, the numerical results are analytically or algebraically confirmed.
The author studies integer S>1 spin chains. He extends the Kennedy-Tasaki nonlocal unitary transformation for S=1 to arbitrary integer S. He shows the main results of Kennedy and Tasaki (1992) are maintained for S>1: Heisenberg-type Hamiltonians are transformed to Hamiltonians of nearest-neighbour interactions with Z 2 *Z 2 symmetry, and the den Nijs-Rommelse string observables are transformed to the ferromagnetic correlation observables. He asserts that in general values of integer S there exist several phases with the hidden Z 2 *Z 2 symmetry breaking. The den Nijs-Rommelse string order parameters, which measure the hidden Z 2 *Z 2 symmetry breaking, are calculated explicitly for several variants of the VBS-type states. In the standard VBS state, the hidden Z 2 *Z 2 symmetry breaks down when S is odd but remains unbroken when S is even. His results for partially dimerized VBS states suggest that the hidden Z 2 *Z 2 symmetry breaking can be used to detect the successive dimerization transitions predicted by Affleck and Haldane (1987). Some new anisotropic VBS-type states are investigated. The result suggests that there are successive phase transitions when he increases the uniaxial anisotropy in a Heisenberg-type model. Other new VBS-type states with long-range order are considered, and their relevance to the phase diagram of the Heisenberg XXZ model and the magnetization process of antiferromagnets is investigated. He introduces an extended string order parameter which possesses a characteristic behavior in the partially dimerized VBS states.
One-dimensional valence bond solid (VBS) states represent the simplest symmetry-protected topological phases. We show that their ground state entanglement spectrum contains both topological and nontopological structures. For the SO(3)\text{SO}(3)SO(3) symmetric VBS states with odd-integer spins, the twofold topological degeneracy is associated with an underlying ${Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}$ symmetry that protects the corresponding topological phase. In general, for the SO(2S+1)\text{SO}(2S+1)SO(2S+1) symmetric VBS states with integer spins SSS, the corresponding protecting symmetry is identified as the ${({Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2})}^{S}$ symmetry, yielding the 2S{2}^{S}2S-fold topological degeneracy. The topological degeneracy and associated protecting symmetry can be identified by a nonlocal unitary transformation, which changes the topological order of the VBS states into conventional ferromagnetic order.
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.