On the singularity structure of the 2D Ising model susceptibility.

*(English)* Zbl 0936.82006
Summary: Some simplifications of the integrals ${\chi}^{(2n+1)}$, derived by *T. T. Wu* [Phys. Rev. B (3) 13, 316-374 (1976)] that contribute to the zero field susceptibility of the 2D square lattice Ising model are reported. In particular, several alternate expressions for the integrands in ${\chi}^{(2n+1)}$ are determined which greatly facilitate both the generation of high-temperature series and analytical analysis. One can show that as series, ${\chi}^{(2n+1)}={2}^{2n}{(s/2)}^{4n(n+1)}(1+O\left(s\right))$ where $s$ is the high-temperature variable $sinh\left(2K\right)$ with $K$ the conventional normalized inverse temperature. Analysis of the integrals near symmetry points of the integrands shows that ${\chi}^{(2n+1)}\left(s\right)$ is singular on the unit circle at ${s}_{k\ell}=exp\left(i{\theta}_{k\ell}\right)$ where $2cos\left({\theta}_{k\ell}\right)=cos(2\pi k/(2n+1))+cos(2\pi \ell /(2n+1))$, $-n\le k$, $\ell \le n$. The singularities, ${\theta}_{k\ell}=0$ excepted, are logarithmic branch points of order ${\epsilon}^{2n(n+1)-1}ln\left(\epsilon \right)$ with $\epsilon =1-s/{s}_{k\ell}$. There is numerical evidence from series that these van Hove points, in addition to the known points at $s=\pm 1$ and $\pm i$, exhaust the singularities on the unit circle. Barring cancellation from extra (unobserved) singularities one can conclude that $\left[s\right]=1$ is a natural boundary for the susceptibility.

##### MSC:

82B20 | Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs |

82B23 | Exactly solvable models; Bethe ansatz |