Some new results on Yang-Lee zeros of the Ising model partition function

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 −2 βH ) lie on the unit circle lhaiμlhai = 1 for a wider range of K nn′ = βJ nn′ than the range K nn′ ≥ 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 J nn′ ≥ 0. For the case of uniform couplings K nn′ = K, we show that these zeros lie on the unit circle lhaiμlhai = 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,l ≤ u ≤ 0 on the triangular lattice, where u = z 2 = e −4 K , 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 lhaiμlhai = 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.