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

We prove that for the Ising model on a lattice of dimensionality d greater than or equal to 2, the zeros of the partition function Z in the complex mu plane (where mu = e(-2 beta H)) lie on the unit circle \mu\ = 1 for a wider range of K-nn' = beta J(nn') than the range K-nn' greater than or equal to 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') greater than or equal to 0. For the case of uniform couplings K-nn' = K, we show that these zeros lie on the unit circle \mu\ = 1 not just for the Yang-Lee range 0 less than or equal to u less than or equal to 1, but also for (i) -u(c,sq) less than or equal to u less than or equal to 0 on the square lattice, and (ii) -u(c,t) less than or equal to u less than or equal to 0 on th 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 x 12(2), and 4 x 8(2) lattices we prove an exact symmetry of the reduced partition functions, Z(r)(z, -mu) = Z(r)(-z, mu). This proves that the zeros of Z for these lattices lie on \mu\ = 1 for -1 less than or equal to z less than or equal to 0 as well as the Yang-Lee range 0 less than or equal to z less than or equal to 1. Finally, we report some new results on the patterns of zeros for values of u or z outside these ranges.