Square lattice Ising model susceptibility: connection matrices and singular behavior of ${\chi }^{(3)}$ and ${\chi }^{(4)}$

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 (${\chi }^{(3)}$ and ${\chi }^{(4)}$) to the magnetic susceptibility of square lattice Ising model. We deduce all the critical behaviors of the solutions ${\chi }^{(3)}$ and ${\chi }^{(4)}$, 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 ${\chi }^{(3)}$ are not singularities of the particular solution ${\chi }^{(3)}$ itself. We use the previous connection matrices to get the exact expressions of all the monodromy matrices of the Fuchsian differential equation for ${\chi }^{(3)}$ (and ${\chi }^{(4)}$) expressed in the same basis of solutions. These monodromy matrices are the generators of the differential Galois group of the Fuchsian differential equations for ${\chi }^{(3)}$ (and ${\chi }^{(4)}$), 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 $\chi$ at high temperature (resp. low temperature) and the first two terms ${\chi }^{(1)}$ and ${\chi }^{(3)}$ (resp. ${\chi }^{(2)}$ and ${\chi }^{(4)}$) .

38 pages, no figure Note added in the proofs

References in corpus (1)

• Ising model susceptibility: Fuchsian differential equation for ${\chi }^{(4)}$ and its factorization properties

Cited by in corpus (28)

• Experimental mathematics on the magnetic susceptibility of the square lattice Ising model
• Ising n-fold integrals as diagonals of rational functions and integrality of series expansions
• High order Fuchsian equations for the square lattice Ising model: ${\tilde{\chi }}^{(5)}$
• The importance of the Ising model
• The diagonal Ising susceptibility
• The Ising model: from elliptic curves to modular forms and Calabi-Yau equations
• Singularities of $n$-fold integrals of the Ising class and the theory of elliptic curves
• Holonomy of the Ising model form factors
• Diagonals of rational functions and selected differential Galois groups
• Landau singularities and singularities of holonomic integrals of the Ising class
• Holonomic functions of several complex variables and singularities of anisotropic Ising n-fold integrals
• Painleve versus Fuchs
• Ising n-fold integrals as diagonals of rational functions and integrality of series expansions: integrality versus modularity
• Diagonals of rational functions, pullbacked 2F1 hypergeometric functions and modular forms (unabrigded version)
• Square lattice Ising model ${\tilde{\chi }}^{(5)}$ ODE in exact arithmetic
• Selected non-holonomic functions in lattice statistical mechanics and enumerative combinatorics
• The anisotropic Ising correlations as elliptic integrals: duality and differential equations
• Canonical decomposition of linear differential operators with selected differential Galois groups
• Is the full susceptibility of the square-lattice Ising model a differentially algebraic function?
• From Holonomy of the Ising Model Form Factors to n-Fold Integrals and the Theory of Elliptic Curve
• Scaling functions in the square Ising model
• Renormalization, isogenies and rational symmetries of differential equations
• Beyond series expansions: mathematical structures for the susceptibility of the square lattice Ising model
• Differential algebra on lattice Green functions and Calabi-Yau operators (unabridged version)
• The Ising model and Special Geometries
• Magnetic susceptibility of the square lattice Ising model
• Errata and Addenda to Mathematical Constants
• Automata and the susceptibility of the square lattice Ising model modulo powers of primes