### Abstract

Haldane predicted that the isotropic quantum Heisenberg spin chain is in a "massive" phase if the spin is integral. The first rigorous example of an isotropic model in such a phase is presented. The Hamiltonian has an exact SO(3) symmetry and is translationally invariant, but we prove the model has a unique ground state, a gap in the spectrum of the Hamiltonian immediately above the ground state and exponential decay of the correlation functions in the ground state. Models in two and higher dimension which are expected to have the same properties are also presented. For these models we construct an exact ground state, and for some of them we prove that the two-point function decays exponentially in this ground state. In all these models exact ground states are constructed by using valence bonds.

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

Pages (from-to) | 477-528 |

Number of pages | 52 |

Journal | Communications in Mathematical Physics |

Volume | 115 |

Issue number | 3 |

DOIs | |

State | Published - Sep 1988 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Physics and Astronomy(all)
- Mathematical Physics

### Cite this

*Communications in Mathematical Physics*,

*115*(3), 477-528. https://doi.org/10.1007/BF01218021

**Valence bond ground states in isotropic quantum antiferromagnets.** / Affleck, Ian; Kennedy, Thomas G; Lieb, Elliott H.; Tasaki, Hal.

Research output: Contribution to journal › Article

*Communications in Mathematical Physics*, vol. 115, no. 3, pp. 477-528. https://doi.org/10.1007/BF01218021

}

TY - JOUR

T1 - Valence bond ground states in isotropic quantum antiferromagnets

AU - Affleck, Ian

AU - Kennedy, Thomas G

AU - Lieb, Elliott H.

AU - Tasaki, Hal

PY - 1988/9

Y1 - 1988/9

N2 - Haldane predicted that the isotropic quantum Heisenberg spin chain is in a "massive" phase if the spin is integral. The first rigorous example of an isotropic model in such a phase is presented. The Hamiltonian has an exact SO(3) symmetry and is translationally invariant, but we prove the model has a unique ground state, a gap in the spectrum of the Hamiltonian immediately above the ground state and exponential decay of the correlation functions in the ground state. Models in two and higher dimension which are expected to have the same properties are also presented. For these models we construct an exact ground state, and for some of them we prove that the two-point function decays exponentially in this ground state. In all these models exact ground states are constructed by using valence bonds.

AB - Haldane predicted that the isotropic quantum Heisenberg spin chain is in a "massive" phase if the spin is integral. The first rigorous example of an isotropic model in such a phase is presented. The Hamiltonian has an exact SO(3) symmetry and is translationally invariant, but we prove the model has a unique ground state, a gap in the spectrum of the Hamiltonian immediately above the ground state and exponential decay of the correlation functions in the ground state. Models in two and higher dimension which are expected to have the same properties are also presented. For these models we construct an exact ground state, and for some of them we prove that the two-point function decays exponentially in this ground state. In all these models exact ground states are constructed by using valence bonds.

UR - http://www.scopus.com/inward/record.url?scp=0009504187&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0009504187&partnerID=8YFLogxK

U2 - 10.1007/BF01218021

DO - 10.1007/BF01218021

M3 - Article

AN - SCOPUS:0009504187

VL - 115

SP - 477

EP - 528

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 3

ER -