# Experimental mathematics on the magnetic susceptibility of the square lattice Ising model - Mathematical Physics

Abstract: We calculate very long low- and high-temperature series for thesusceptibility $\chi$ of the square lattice Ising model as well as very longseries for the five-particle contribution $\chi^{5}$ and six-particlecontribution $\chi^{6}$. These calculations have been made possible by theuse of highly optimized polynomial time modular algorithms and a total of morethan 150000 CPU hours on computer clusters. For $\chi^{5}$ 10000 terms of theseries are calculated {\it modulo} a single prime, and have been used to findthe linear ODE satisfied by $\chi^{5}$ {\it modulo} a prime.A diff-Pad\-e analysis of 2000 terms series for $\chi^{5}$ and $\chi^{6}$confirms to a very high degree of confidence previous conjectures about thelocation and strength of the singularities of the $n$-particle components ofthe susceptibility, up to a small set of additional- singularities. We findthe presence of singularities at $w=1-2$ for the linear ODE of $\chi^{5}$,and $w^2= 1-8$ for the ODE of $\chi^{6}$, which are {\it not} singularitiesof the physical- $\chi^{5}$ and $\chi^{6},$ that is to say theseries-solutions of the ODE-s which are analytic at $w =0$.Furthermore, analysis of the long series for $\chi^{5}$ and $\chi^{6}$combined with the corresponding long series for the full susceptibility $\chi$yields previously conjectured singularities in some $\chi^{n}$, $n \ge 7$.We also present a mechanism of resummation of the logarithmic singularitiesof the $\chi^{n}$ leading to the known power-law critical behaviour occurringin the full $\chi$, and perform a power spectrum analysis giving strongarguments in favor of the existence of a natural boundary for the fullsusceptibility $\chi$.

Author: S. Boukraa, A. J. Guttmann, S. Hassani, I. Jensen, J.-M. Maillard, B. Nickel, N. Zenine

Source: https://arxiv.org/