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On the singularity structure of the 2D Ising model susceptibility. (English) Zbl 0936.82006
Summary: Some simplifications of the integrals χ (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 χ (2n+1) are determined which greatly facilitate both the generation of high-temperature series and analytical analysis. One can show that as series, χ (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 χ (2n+1) (s) is singular on the unit circle at s k =exp(iθ k ) where 2cos(θ k )=cos(2πk/(2n+1))+cos(2π/(2n+1)), -nk, n. The singularities, θ k =0 excepted, are logarithmic branch points of order ε 2n(n+1)-1 ln(ε) with ε=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.

MSC:
82B20Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs
82B23Exactly solvable models; Bethe ansatz