The Ising model was introduced by Lenz in 1920 and solved in one dimension by Ising in 1925.It is defined by placing "spin" variables which take on the values on the sites of a latticeand there an interaction energy between nearest neighbor spins of if the spins have the same value and if the spins have opposite values. Ising (Z. Physik, 31, 253, 1925) introduced a model consisting of a lattice of \spin" variables s. i, which can only take the values +1 (") and 1 (#). The sum over the full configuration space spans over exactly states, because each spin can only have 2 possible values. The two-dimensional Ising model 1/2 2.1 An exactly solvable model of phase transition 2.1.1 Introduction One of the main concerns in Statistical Mechanics is the study of phase transitions, when the state of a system changes dramatically. (Onsager (1944)) An analytical solution for the general case for $${\displaystyle H\neq 0}$$ has yet to be found. 2D Ising Correlation Function The spin-spin correlation functions for the two-dimensional Ising model is known exactly at zero external field. The relevant Grassmann action is quadratic, so that the solution can… In statistical mechanics, the two-dimensional square lattice Ising model is a simple lattice model of interacting magnetic spins. The partition function of the 2-D Ising model. The model is notable for having nontrivial interactions, yet having an analytical solution. Because of its simplicity it is possible to solve it analytically in 1 and 2 dimensions, for it is not solved yet in 3 or higher dimensions. The Ising model is a very simple model to describe magnetism in solid state bodies. He used Grassmann variables to formulate the problem in terms of a free-fermion model, via the fermionic path integral approach. As a topic, it is chosen the 2D Ising model to discuss its physical importance using adequate mathematical formalisms. Moreover, since the sum is finite (for finite ), we can write the -sum as iterated sums, … On a rectangular lattice of rows and columnswith periodic boundary conditions in both directionsthis gives a total interaction energy of where is the spin in row and column the last term i… It is expressed in terms of integrals of Painlevé functions which, while of fundamental importance in many fields of physics, are not provided in most software environments. 1. ig) = J X. hi;ji. s. Every spin interacts with its nearest neighbors (2 in 1D) as well as with an external magnetic eld h. The Hamiltonian1of the Ising model is H(fs. The model was solved by Lars Onsager for the special case that the external magnetic field H = 0. Introduction In 1980 Stuart Samuel gave what I consider to be one of the most elegant exact solutions of the 2D Ising model.