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 $\chi$ of the square lattice Ising model as well as very long series for the five-particle contribution ${\chi }^{(5)}$ and six-particle contribution ${\chi }^{(6)}$. 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 ${\chi }^{(5)}$ 10000 terms of the series are calculated {\it modulo} a single prime, and have been used to find the linear ODE satisfied by ${\chi }^{(5)}$ {\it modulo} a prime. A diff-Padé analysis of 2000 terms series for ${\chi }^{(5)}$ and ${\chi }^{(6)}$ 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 ${\chi }^{(5)}$, and $w^2=1/8$ for the ODE of ${\chi }^{(6)}$, which are {\it not} singularities of the ``physical'' ${\chi }^{(5)}$ and ${\chi }^{(6)},$ that is to say the series-solutions of the ODE's which are analytic at $w=0$. Furthermore, analysis of the long series for ${\chi }^{(5)}$ (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 singularities of the ${\chi }^{(n)}$ leading to the known power-law critical behaviour occurring in the full $\chi$, and perform a power spectrum analysis giving strong arguments in favor of the existence of a natural boundary for the full susceptibility $\chi$.

54 pages, 2 figures