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arXiv · 1002.2480

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

Vladimir V. Mangazeev·Anthony J. Guttmann
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Abstract

We derive a Toda-type recurrence relation, in both high and low temperature regimes, for the $λ$ - extended diagonal correlation functions $C(N,N;λ)$ 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 $λ$-extended counterparts. We also conjecture a closed form expression for the simplest off-diagonal case $C^{\pm}(0,1;λ)$ 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.

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Vladimir V. Mangazeev, Anthony J. Guttmann. 2010-04-28. Form factor expansions in the 2D Ising model and Painlevé VI. https://doi.org/10.1016/j.nuclphysb.2010.05.021

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