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 {\it 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, $\chi^{(2n+1)}= 2^{2n} (s/2)^{4n(n+1)} (1+O(s))$ where $s$ is the high-temperature variable $\sinh(2K)$ with $K$ the conventional normalized inverse temperature. Analysis of the integrals near symmetry points of the integrands shows that $\chi^{(2n+1)} (s)$ is singular on the unit circle at $s_{k\ell}= \exp(i\theta_{k\ell})$ where $2\cos(\theta_{k\ell})= \cos(2\pi k/(2n+1))+ \cos(2\pi\ell/ (2n+1))$, $-n\leq k$, $\ell\leq n$. The singularities, $\theta_{k\ell}= 0$ excepted, are logarithmic branch points of order $\varepsilon^{2n(n+1)-1} \ln(\varepsilon)$ with $\varepsilon= 1-s/s_{k\ell}$. 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 $[s]= 1$ 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 |