## Abstract

We prove that for the Ising model on a lattice of dimensionality d ≥ 2, the zeros of the partition function Z in the complex μ plane (where μ = e^{-2βH}) lie on the unit circle |μ| = 1 for a wider range of Knn′ = βJnn′ than the range Knn′ ≥ 0 assumed in the premise of the Yang-Lee circle theorem. This range includes complex temperatures, and we show that it is lattice-dependent. Our results thus complement the Yang-Lee theorem, which applies for any d and any lattice if Jnn′ ≥ 0. For the case of uniform couplings Knn′ = K, we show that these zeros lie on the unit circle |μ| = 1 not just for the Yang-Lee range 0 ≤ u ≤1, but also for (i) -u_{c,sq} ≤ u ≤ 0 on the square lattice, and (ii) -u_{c,t} ≤ u ≤ 0 on the triangular lattice, where u = z^{2} = e^{-4K}, u_{c,sq} = 3 - 2^{3/2}, and u_{c,t} = 1/3. For the honeycomb, 3 × 12^{2}, and 4 × 8^{2} lattices we prove an exact symmetry of the reduced partition functions, Z_{r}(z, -μ) = Z_{r}(-z,μ). This proves that the zeros of Z for these lattices lie on |μ| = 1 for -1 ≤ z ≤ 0 as well as the Yang-Lee range 0 ≤ z ≤ 1. Finally, we report some new results on the patterns of zeros for values of u or z outside these ranges.

Original language | English (US) |
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Pages (from-to) | 271-279 |

Number of pages | 9 |

Journal | Physics Letters, Section A: General, Atomic and Solid State Physics |

Volume | 215 |

Issue number | 5-6 |

DOIs | |

State | Published - Jun 10 1996 |

Externally published | Yes |

## All Science Journal Classification (ASJC) codes

- Physics and Astronomy(all)