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

Abstract

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 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 $χ$.