Experimental mathematics on the magnetic susceptibility of the square lattice Ising model

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dc.creatorBoukraa, S.
dc.creatorGuttmann, A. J.
dc.creatorHassani, S.
dc.creatorJensen, I.
dc.creatorMaillard, J. -M.
dc.creatorNickel, B.
dc.creatorZenine, N.
dc.date2008-08-06
dc.date.accessioned2026-07-07T11:45:21Z
dc.date.available2026-07-07T11:45:21Z
dc.descriptionWe 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 150000 CPU hours on computer clusters. For $χ^{(5)}$ 10000 terms of the series are calculated {\it modulo} a single prime, and have been used to find the linear ODE satisfied by $χ^{(5)}$ {\it modulo} a prime. A diff-Padé analysis of 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. 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 {\it 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 $χ^{(n)}$, $n \ge 7$. 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 favor of the existence of a natural boundary for the full susceptibility $χ$.
dc.description54 pages, 2 figures
dc.identifierhttps://arxiv.org/abs/0808.0763
dc.identifierhttp://arxiv.org/abs/0808.0763
dc.identifierJ.Phys.A41:455202,2008
dc.identifierdoi:10.1088/1751-8113/41/45/455202
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/201849
dc.subjectMathematical Physics
dc.subjectStatistical Mechanics
dc.subjectHigh Energy Physics - Theory
dc.subject34M55, 47E05, 81Qxx, 32G34, 34Lxx, 34Mxx, 14Kxx
dc.titleExperimental mathematics on the magnetic susceptibility of the square lattice Ising model
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