PHYSICS LETTERS A ELSEVIER Physics Letters A 215 ( 1996) 27 I-279 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 2 2, the zeros of the partition function 2 in the complex p plane (where,u = e-2PH) lie on the unit circle j,~l = 1 for a wider range of K,, / = /3J,,,*t than the range K,,,,, 3 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,,,P> 0. For the case of uniform couplings K,,, = K, we show that these zeros lie on the unit circle 1~1 = 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)-uuc,, < u < 0 on the triangular lattice, where u = z2 = e-4K, L+ = 3- 2312, and u~, ~ = l/3. For the honeycomb, 3 x 12*, and 4 x 8 * lattices we prove an exact symmetry of the reduced partition functions, & ( z,-_ICL) = Zr (--I, /A). This proves that the zeros of Z for these lattices lie on 1~1 = I for-1 < z 6 0 as well as the Yang-Lee range 0 < z 6 1. Finally, we report some new results on the patterns of zeros for values of u or z, outside these ranges. The Ising model has long served as a simple proto-type of a statistical mechanical system which (for d greater than the lower critical dimensionality d1,c.d. = 1) undergoes a second-order phase transition wit