The Ising correlation C(M, N) for ν = −k

• Published in: Journal of physics. A, Mathematical and theoretical Mathematical and theoretical, 2020-11, Vol.53 (46), p.465202
• Main Authors: Boukraa, S, Maillard, J-M, McCoy, B M
• Format: Article
• Language: English
• Subjects: Ising correlation functions; Okamoto parameters; sigma form of Painlevé VI
• ISSN: 1751-8113; 1751-8121
• DOI: 10.1088/1751-8121/abbb61

We present Painlevé VI sigma form equations for the general Ising low and high temperature two-point correlation functions C(M, N) with M ⩽ N in the special case ν = −k where ν = sinh 2Eh/kBT/sinh 2Ev/kBT. More specifically four different non-linear ODEs depending explicitly on the two integers M and N emerge: these four non-linear ODEs correspond to distinguish respectively low and high temperature, together with M + N even or odd. These four different non-linear ODEs are also valid for M ⩾ N when ν = −1/k. For the low-temperature row correlation functions C(0, N) with N odd, we exhibit again for this selected ν = −k condition, a remarkable phenomenon of a Painlevé VI sigma function being the sum of four Painlevé VI sigma functions having the same Okamoto parameters. We show in this ν = −k case for T < Tc and also T > Tc, that C(M, N) with M ⩽ N is given as an N × N Toeplitz determinant.