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## Ising model in a field

The final piece of work in the Ising model to be mentioned is what happens to the two point function when a small magnetic field is put the system for T<Tc. At H=0 for T<Tc the two point function has the important property that it couples only to states with an even number of particles and thus, in particular the leading singularity in the Fourier transform is not a single particle pole but rather a two particle cut. In 1978 [16], as an application of our explicit formulas for the n spin correlation functions we did a (singular) perturbation computation to see what happens when in terms of the scaled variable h=H/|T-Tc|15/8 a small value of h is applied to the system. We found that the two particle cut breaks up into an infinite number of poles which are given by the zeroes of Airy functions. These poles are exactly at the positions of the bound states of a linear potential and are immediately interpretable as a weak confinement of the particles which are free at H=0. This is perhaps the earliest explicit computation where confinement is seen. From this result it was natural to conjecture that as we take the Ising model from T>TcH=0 to T<TcH=0 that as h increases from to that bound states emerge from the two particle cut and that as we further proceed from T=TcH>0 down to T<TcH=0 that bound states continue to emerge until at H=0 an infinite number of bound states have emerged and formed a two particle cut. What this picture does not indicate is the remarkable result found 10 years later by A. Zamolodchikov [22]that at T=TcH=0 the problem can again be studied exactly. This totally unexpected result will be discussed (if the abstract is correct) be Sasha in the next presentation.

Next: From Ising to integrable Up: The Ising Model Previous: Painlevé Functions and difference
Barry McCoy
3/29/1999