### Abstract

For a large class of one-dimensional spin-1 Hamiltonians with open boundary conditions, we show that there is a unitary transformation for which the off-diagonal matrix elements of the transformed Hamiltonian are non-positive. We use this to show that the ground state of a finite chain is at most fourfold degenerate, and that the expectation of the string observable of den Nijs and Rommelse in the ground state is bounded below by the expectation of the usual Neel order parameter. (This was proved for a smaller class of Hamiltonians by Kennedy and Tasaki.) The class of Hamiltonians to which our results apply include the general isotropic Hamiltonian Σ_{i}[S_{i} · S_{i+1} - β(S_{i} · S_{i+1})^{2}] for β > -1. For the usual Heisenberg Hamiltonian the transformed Hamiltonian is - Σ_{i}T_{i} · T_{i+1} where the operators T = (T^{x}, T^{y}, T^{z}) satisfy anticommutation relations like {T^{x}, T^{y}} = T^{z}. We can also use this transformation to obtain variational bounds on the ground-state energy. The transformation used here is closely related to the unitary operator introduced by Kennedy and Tasaki.

Original language | English (US) |
---|---|

Pages (from-to) | 8015-8022 |

Number of pages | 8 |

Journal | Journal of Physics Condensed Matter |

Volume | 6 |

Issue number | 39 |

DOIs | |

State | Published - 1994 |

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### ASJC Scopus subject areas

- Condensed Matter Physics
- Electronic, Optical and Magnetic Materials

### Cite this

**Non-positive matrix elements for Hamiltonians of spin-1 chains.** / Kennedy, Thomas G.

Research output: Contribution to journal › Article

*Journal of Physics Condensed Matter*, vol. 6, no. 39, pp. 8015-8022. https://doi.org/10.1088/0953-8984/6/39/020

}

TY - JOUR

T1 - Non-positive matrix elements for Hamiltonians of spin-1 chains

AU - Kennedy, Thomas G

PY - 1994

Y1 - 1994

N2 - For a large class of one-dimensional spin-1 Hamiltonians with open boundary conditions, we show that there is a unitary transformation for which the off-diagonal matrix elements of the transformed Hamiltonian are non-positive. We use this to show that the ground state of a finite chain is at most fourfold degenerate, and that the expectation of the string observable of den Nijs and Rommelse in the ground state is bounded below by the expectation of the usual Neel order parameter. (This was proved for a smaller class of Hamiltonians by Kennedy and Tasaki.) The class of Hamiltonians to which our results apply include the general isotropic Hamiltonian Σi[Si · Si+1 - β(Si · Si+1)2] for β > -1. For the usual Heisenberg Hamiltonian the transformed Hamiltonian is - ΣiTi · Ti+1 where the operators T = (Tx, Ty, Tz) satisfy anticommutation relations like {Tx, Ty} = Tz. We can also use this transformation to obtain variational bounds on the ground-state energy. The transformation used here is closely related to the unitary operator introduced by Kennedy and Tasaki.

AB - For a large class of one-dimensional spin-1 Hamiltonians with open boundary conditions, we show that there is a unitary transformation for which the off-diagonal matrix elements of the transformed Hamiltonian are non-positive. We use this to show that the ground state of a finite chain is at most fourfold degenerate, and that the expectation of the string observable of den Nijs and Rommelse in the ground state is bounded below by the expectation of the usual Neel order parameter. (This was proved for a smaller class of Hamiltonians by Kennedy and Tasaki.) The class of Hamiltonians to which our results apply include the general isotropic Hamiltonian Σi[Si · Si+1 - β(Si · Si+1)2] for β > -1. For the usual Heisenberg Hamiltonian the transformed Hamiltonian is - ΣiTi · Ti+1 where the operators T = (Tx, Ty, Tz) satisfy anticommutation relations like {Tx, Ty} = Tz. We can also use this transformation to obtain variational bounds on the ground-state energy. The transformation used here is closely related to the unitary operator introduced by Kennedy and Tasaki.

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U2 - 10.1088/0953-8984/6/39/020

DO - 10.1088/0953-8984/6/39/020

M3 - Article

VL - 6

SP - 8015

EP - 8022

JO - Journal of Physics Condensed Matter

JF - Journal of Physics Condensed Matter

SN - 0953-8984

IS - 39

ER -