Square lattice Ising model susceptibility: connection matrices and singular behaviour of χ⁽³⁾and χ⁽⁴⁾

Abstract

Abstract. 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 the square lattice Ising model. We deduce all the critical behavior 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 singularities of the Fuchsian ODE associated with 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) ) .

Landau singularities and singularities of holonomic integrals of the Ising class

“…This connection matrix method [8] was used to find the singular behavior of χ (3) and χ (4) , the third and fourth "particle" contribution to the Ising magnetic susceptibility [8]. Let us sketch the method.…”

“…For the three-particle contribution χ (3) , we have shown [8] that the two quadratic roots of 1 + 3w + 4w 2 = 0, singularities of the linear ODE, are actually absent in the multiple integral defining χ (3) written as a function of the self-dual variable w ‡.…”

“…In previous works, we have introduced a "connection matrix method" [8] to determine if a singularity of the linear ODE is actually a singularity occurring in the multiple integral. For the three-particle contribution χ (3) , we have shown [8] that the two quadratic roots of 1 + 3w + 4w 2 = 0, singularities of the linear ODE, are actually absent in the multiple integral defining χ (3) written as a function of the self-dual variable w ‡.…”

“…In our holonomic approach, the singularities of the multiple integrals are, thus, obtained in two steps: find the linear differential equation, then use our "connection method" [8] in order to analyze the singularities of the solutions of this ODE and, especially, the singularities of that particular solution corresponding to the multiple integral. These computations may be rather cumbersome, depending on the order of the linear ODE, on the number of singularities, and their distribution in the complex plane of the variable.…”

“…Recall that small w corresponds to small values of s, as well as large values of s. If successful, the Landau conditions will give the singularities of these integrals bypassing the linear ODE search and the singularity analyzis from our connection method [8], possibly providing a deeper insight on the natural boundary problem of the susceptibility.…”