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 $K_{n{n}^{\prime }}=\beta J_{n{n}^{\prime }}$ than the range $K_{n{n}^{\prime }}\ge 0$ 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 $J_{n{n}^{\prime }}\ge 0$. For the case of uniform couplings $K_{n{n}^{\prime }}=K$, 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.

10 pages, latex, with separate compressed, uuencoded figures. one typo corrected in abstract

References in corpus (6)

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