We present a simple, but efficient, way to calculate connection matrices between sets of independent local solutions, defined at two neighboring singular points, of Fuchsian differential equations of quite large orders, such as those found for the third and fourth contribution ($χ^{(3)}$ and $χ^{(4)}$) to the magnetic susceptibility of square lattice Ising model. We deduce all the critical behaviors of the solutions $χ^{(3)}$ and $χ^{(4)}$, as well as the asymptotic behavior of the coefficients in the corresponding series expansions. We confirm that the newly found quadratic number singularities of the Fuchsian ODE associated to $χ^{(3)}$ are not singularities of the particular solution $χ^{(3)}$ itself.
We use the previous connection matrices to get the exact expressions of all the monodromy matrices of the Fuchsian differential equation for $χ^{(3)}$ (and $χ^{(4)}$) expressed in the same basis of solutions. These monodromy matrices are the generators of the differential Galois group of the Fuchsian differential equations for $χ^{(3)}$ (and $χ^{(4)}$), whose analysis is just sketched here.
As far as the physics implications of the solutions are concerned, we find challenging qualitative differences when comparing the corrections to scaling for the full susceptibillity $χ$ at high temperature (resp. low temperature) and the first two terms $χ^{(1)}$ and $χ^{(3)}$
(resp. $χ^{(2)}$ and $χ^{(4)}$) .
Energy field of critical Ising model and examples of singular fields in QFT
Université Claude Bernard Lyon 1（里昂第一大学）；CNRS UMR 5208, Institut Camille Jordan（法国国家科学研究中心 5208 联合研究实验室，卡米尔·乔丹研究所）；Institut Universitaire de France (IUF)（法兰西学院）；University of Helsinki（赫尔辛基大学）
