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) -uc,sq ≤ u ≤ 0 on the square lattice, and (ii) -uc,t ≤ u ≤ 0 on the triangular lattice, where u = z2 = e-4K, uc,sq = 3 - 23/2, and uc,t = 1/3. For the honeycomb, 3 × 122, and 4 × 82 lattices we prove an exact symmetry of the reduced partition functions, Zr(z, -μ) = Zr(-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)|
|Number of pages||9|
|Journal||Physics Letters, Section A: General, Atomic and Solid State Physics|
|State||Published - Jun 10 1996|
All Science Journal Classification (ASJC) codes
- Physics and Astronomy(all)