Square lattice Ising model ${\tilde{\chi }}^{(5)}$ ODE in exact arithmetic

Abstract

We obtain in exact arithmetic the order 24 linear differential operator $L_{24}$ and right hand side $E^{(5)}$ of the inhomogeneous equation$L_{24}({\Phi }^{(5)})=E^{(5)}$, where ${\Phi }^{(5)}={\tilde{\chi }}^{(5)}-{\tilde{\chi }}^{(3)}/2+{\tilde{\chi }}^{(1)}/120$ is a linear combination of $n$-particle contributions to the susceptibility of the square lattice Ising model. In Bostan, et al. (J. Phys. A: Math. Theor. {\bf 42}, 275209 (2009)) the operator $L_{24}$ (modulo a prime) was shown to factorize into $L_{12}^{(\mathrm{left})}\cdot L_{12}^{(\mathrm{right})}$; here we prove that no further factorization of the order 12 operator $L_{12}^{(\mathrm{left})}$ is possible. We use the exact ODE to obtain the behaviour of ${\tilde{\chi }}^{(5)}$ at the ferromagnetic critical point and to obtain a limited number of analytic continuations of ${\tilde{\chi }}^{(5)}$ beyond the principal disk defined by its high temperature series. Contrary to a speculation in Boukraa, et al (J. Phys. A: Math. Theor. {\bf 41} 455202 (2008)), we find that ${\tilde{\chi }}^{(5)}$ is singular at $w=1/2$ on an infinite number of branches.

25 pages, 2 figures, IoP style files

References in corpus (2)

• Experimental mathematics on the magnetic susceptibility of the square lattice Ising model
• Globally nilpotent differential operators and the square Ising model

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• Scaling functions in the square Ising model
• The Ising model and Special Geometries
• Magnetic susceptibility of the square lattice Ising model