Spin Models (Ising And Landau)

One of the most interesting, perplexing, and difficult things to understand in modern Physics is Phase transitions. There are many examples of this: transition from solid to liquid to gas, superconductivity, ferromagnetism and paramagnetism. The most difficult part is to model the interactions between the microscopic particles. There are various approaches to this, and it is still a growing area of research, and I will examine a few.

Perhaps one of the most simplistic attempts to model this was done by Landau. This approach involves expanding the Free energy as a function of the parameter (the parameter is the macroscopic quantity that we are interested in which could be Magnetism, superconductivity, etc) and the critical quantity (could be Temperature, or Magnetic field) around the critical point. We then differentiate with respect to the parameter and set it equal to zero to find the minima of the Free energy. There are a few problems with this approach. First, the Free energy is not analytic in terms of the critical quantity. Second, Landau's theory does not account for the dimensionality of the system, and thus incorrectly predicts a transition in one dimension.

In order to obtain a clearer picture of transitions, physicists introduced the model of spins. The most famous of these models is the Ising model. In this model, we take a square network and put N spins on it, each with a value σ = ±1 (this value can represent either an up or down spin for an electron or a magnetic moment that points in either north or south etc.). We then introduce an exterior field H, and naturally the spins will tend orient themselves with the field. Moreover, we must consider the interactions between each spin which is defined by a coupling energy J, which governs how the spins interact with one another.

With these two quantities, J and H, we can write the Hamiltonian for the system. Some interesting conclusions we get from this model are