Kallen-Lehman approach to 3D Ising model

A ”Kallen-Lehman” approach to Ising model, inspired by quantum field theory a la Regge, is proposed. The analogy with the Kallen-Lehman representation leads to a formula for the free-energy of the 3D model with few free parameters which could be matched with the numerical data. The possible application of this scheme to the spin glass case is shortly discussed. Keyword: Regge theory, spin glasses, Ising model. PACS: 12.40.Nn,11.55.Jy, 05.20.-y, 05.70.Fh. Introduction The three dimensional Ising model (henceforth 3DI) is one of the main open problems in field theory and statistical mechanics. A large number of interesting statistical systems near the transition point are described by 3DI and the theoretical methods suitable to deal with such a problem manifest deep connections in various areas of physics ranging from quantum information theory to string theory (to provide with even a partial list of references is really a completely hopeless task, a detailed review-which, by the way, does not cover all the approaches and results on the subject-is [11]). Besides its intrinsic interest is statistical physics, since the formulation of the Svetitsky-Yaffe conjecture [14], it has been widely recognized its role in describing the deconfinement transition in QCD. For this reason, the 3D Ising model is worth to be further investigated. Here, a ”phenomenological” approach is proposed which allows to formally connect the 3DI model to the Ising model in one dimension less. The method is inspired by quantum field theory a la Regge and by the Kallen-Lehman representation. Recently (see [2]) this approach has been able to shed some light on the physical origin of non extensivity. The paper is organized as follows: in the second section the ”Kallen-Lehman interpretation” of the 2D Ising model will be discussed. In the third section the 3D model will be described. In the fourth section the high temperature behavior The meaning of the term ”phenomenological” in this context will be more clear later on. 1 will be analyzed and it will be shortly pointed out how to apply the formalism to the spin glass case. Eventually, some conclusion will be drawn. 1 Ising Kinetic term(s) In this section the main idea of the method are introduced. The partition of functions of Ising models in various dimensions all belong to the following class of partition functions Z(Jab) = ∑