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 $d \ge 2$, 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 $K_{n n'}=\beta J_{nn'}$ than the range $K_{n n'} \ge 0$ 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 $J_{nn'} \ge 0$. 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 $0 \le u \le 1$, but also for (i) $-u_{c,sq} \le u \le 0$ on the square lattice, and (ii) $-u_{c,t} \le u \le 0$ 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, $3 \cdot 12^2$, and $4 \cdot 8^2$ 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 $-1 \le z \le 0$ as well as the Yang-Lee range $0 \le z \le 1$. Finally, we report some new results on the patterns of zeros for values of $u$ or $z$ outside these ranges.