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

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

2010-02-12

arxiv.org/abs/1002.2480v2

Physics

Mathematical Physics

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

Scientific paper

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 Painlev\'e 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 $\l$-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 $\l$-extended case, and facilitate other developments.

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