2008/08/06 by S. Boukraa, A. J. Guttmann, A J Guttmann +9 · 4 citations
Mathematics · Physics and Astronomy · #Markov Chains and Monte Carlo Methods #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #cond-mat.stat-mech #hep-th #math-ph #math.MP #msc:14Kxx #msc:32G34 #msc:34Lxx #msc:34M55 #msc:34Mxx #msc:47E05 #msc:81Qxx
paper · pdf · doi:10.1088/1751-8113/41/45/455202
published as J.Phys.A41:455202,2008 · 54 pages, 2 figures
arxiv created 2008/08/06 · openalex publication_date 2008/10/14 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30
We calculate very long low- and high-temperature series for the susceptibility χ of the square lattice Ising model as well as very long series for the five-particle contribution χ (5) and six-particle contribution χ (6) . These calculations have been made possible by the use of highly optimized polynomial time modular algorithms and a total of more than 150 000 CPU hours on computer clusters. The series for χ (low- and high-temperature regimes), χ (5) and χ (6) are now extended to 2000 terms. In addition, for χ (5) , 10 000 terms of the series are calculated modulo a single prime, and have been used to find the linear ODE satisfied by χ (5) modulo a prime. A diff-Padé analysis of the 2000 terms series for χ (5) and χ (6) confirms to a very high degree of confidence previous conjectures about the location and strength of the singularities of the n -particle components of the susceptibility, up to a small set of 'additional' singularities. The exponents at all the singularities of the Fuchsian linear ODE of χ (5) and the (as yet unknown) ODE of χ (6) are given: they are all rational numbers. We find the presence of singularities at w = 1/2 for the linear ODE of χ (5) , and w 2 = 1/8 for the ODE of χ (6) , which are not singularities of the 'physical' χ (5) and χ (6) , that is to say the series solutions of the ODE's which are analytic at w = 0. Furthermore, analysis of the long series for χ (5) (and χ (6) ) combined with the corresponding long series for the full susceptibility χ yields previously conjectured singularities in some χ ( n ) , n ⩾ 7. The exponents at all these singularities are also seen to be rational numbers. We also present a mechanism of resummation of the logarithmic singularities of the χ ( n ) leading to the known power-law critical behaviour occurring in the full χ, and perform a power spectrum analysis giving strong arguments in favour of the existence of a natural boundary for the full susceptibility χ.