Form factor expansions in the 2D Ising model and Painlevé VI

Abstract

We derive a Toda-type recurrence relation, in both high and low temperature regimes, for the $\lambda$ - extended diagonal correlation functions $C(N,N;\lambda )$ of the two-dimensional Ising model, using an earlier connection between diagonal form factor expansions and tau-functions within Painlevé VI (PVI) theory, originally discovered by Jimbo and Miwa. This greatly simplifies the calculation of the diagonal correlation functions, particularly their $\lambda$-extended counterparts. We also conjecture a closed form expression for the simplest off-diagonal case $C^{\pm }(0,1;\lambda )$ where a connection to PVI theory is not known. Combined with the results for diagonal correlations these give all the initial conditions required for the $ł$-extended version of quadratic difference equations for the correlation functions discovered by McCoy, Perk and Wu. The results obtained here should provide a further potential algorithmic improvement in the $ł$-extended case, and facilitate other developments.

23 pages, references added, introduction extended, abstract modified, misprints corrected

References in corpus (5)

• High order Fuchsian equations for the square lattice Ising model: ${\tilde{\chi }}^{(5)}$
• Holonomy of the Ising model form factors
• Painleve versus Fuchs
• The saga of the Ising susceptibility
• Picard solution of Painlevé VI and related tau-functions

Cited by in corpus (7)

• Exact strong coupling results in $\mathcal{N}$=2 $Sp(2N)$ superconformal gauge theory from localization
• Resurgence, Painleve Equations and Conformal Blocks
• The saga of the Ising susceptibility
• The anisotropic Ising correlations as elliptic integrals: duality and differential equations
• Connection formulas for the lambda generalized Ising correlation functions
• Integrable Differential Systems for Deformed Laguerre-Hahn Orthogonal Polynomials
• On General-n Coefficients in Series Expansions for Row Spin-Spin Correlation Functions in the Two-Dimensional Ising Model