### Abstract

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.

Original language | English |
---|---|

Article number | 255303 |

Journal | Journal of Physics A: Mathematical and Theoretical |

Volume | 43 |

Issue number | 25 |

DOIs | |

Publication status | Published - 2010 |

Externally published | Yes |

### Fingerprint

### ASJC Scopus subject areas

- Mathematical Physics
- Physics and Astronomy(all)
- Statistical and Nonlinear Physics
- Modelling and Simulation
- Statistics and Probability

### Cite this

*Journal of Physics A: Mathematical and Theoretical*,

*43*(25), [255303]. https://doi.org/10.1088/1751-8113/43/25/255303

**Entanglement in valence-bond-solid states on symmetric graphs.** / Katsura, Hosho; Kawashima, Naoki; Kirillov, Anatol N.; Korepin, Vladimir E.; Tanaka, Shu.

Research output: Contribution to journal › Article

*Journal of Physics A: Mathematical and Theoretical*, vol. 43, no. 25, 255303. https://doi.org/10.1088/1751-8113/43/25/255303

}

TY - JOUR

T1 - Entanglement in valence-bond-solid states on symmetric graphs

AU - Katsura, Hosho

AU - Kawashima, Naoki

AU - Kirillov, Anatol N.

AU - Korepin, Vladimir E.

AU - Tanaka, Shu

PY - 2010

Y1 - 2010

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=77952992906&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77952992906&partnerID=8YFLogxK

U2 - 10.1088/1751-8113/43/25/255303

DO - 10.1088/1751-8113/43/25/255303

M3 - Article

VL - 43

JO - Journal of Physics A: Mathematical and Theoretical

JF - Journal of Physics A: Mathematical and Theoretical

SN - 1751-8113

IS - 25

M1 - 255303

ER -