On the singularity structure of the 2D Ising model susceptibility

Some simplifications of the integrals ⁽²ⁿ⁺¹⁾, derived by Wu et al (1976 13 316), 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 ⁽²ⁿ⁺¹⁾ are determined which greatly facilitate both the generation of high-temperature series and analytical analysis. One can show that as series, ⁽²ⁿ⁺¹⁾ = 2²ⁿ(s/2)^4n(n+1)(1+O(s)) where s is the high-temperature variable sin(2K) with K the conventional normalized inverse temperature. Analysis of the integrals near symmetry points of the integrands shows that ⁽²ⁿ⁺¹⁾(s) is singular on the unit circle at s_k = exp(i_k) where 2cos(_k) = cos(2 k/(2n+1))+cos(2/(2n+1)), -n k, n. The singularities, _k = 0 excepted, are logarithmic branch points of order ^2n(n+1)-1ln() with = 1-s/s_k. There is numerical evidence from series that these van Hove points, in addition to the known points at s = ±1 and ±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.