Some New Results on Yang-Lee Zeros of the Ising Model Partition Function

TL;DR

This paper extends the understanding of Yang-Lee zeros of the Ising model, showing they lie on the unit circle under broader conditions, including complex temperatures and various lattice types, thus complementing the classical Yang-Lee theorem.

Contribution

It proves that the zeros of the Ising model's partition function lie on the unit circle for a wider range of parameters and lattice types, including complex temperatures and specific lattice symmetries.

Findings

Zeros lie on the unit circle for complex temperatures and various lattices.

Extended the range of parameters where zeros are on the unit circle beyond the classical theorem.

Identified symmetry properties of partition functions for certain lattices.

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.