The mean models used to describe magnetism: Ising and Heisenberg models (Other model: XY model)

The Heisenberg model of magnetism

 The Heisenberg model occupies a somewhat uncomfortable position in solid state and mathematical physics. Easy to state and widely applicable, it is also extremely difficult to analyze. The interaction model which became known as the Heisenberg model was discovered by Werner Heisenberg and, nearly simultaneously, by P. A. M. Dirac, in 1926. Further work over the following decade established the Heisenberg model, due to its versatility, to be “the fundamental object of study of the theory of magnetism” [13, p. 40]. Since then, much work in theoretical statistical mechanics, solid state physics, and mathematical physics alike has been concentrated on understanding this model. Despite this effort, the Heisenberg model remains largely intractable, its predicted energy levels and eigenfunctions poorly understood, especially in a three-dimensional setting.

The Heisenberg model can be used to describe magnetically ordered solids, in which internal magnetic interactions cause individual magnetic ions to possess nonzero magnetic moments below some critical temperature. Magnetically ordered solids separate broadly into three groups: ferromagnets, antiferromagnets, and ferrimagnets. In ferromagnets, on which we will concentrate, the local moments are energetically preferred to be all aligned in a particular direction, the direction of spontaneous magnetization, so that the solid as a whole has a nonzero spontaneous magnetic moment. In antiferromagnets, adjacent local moments prefer to be antialigned; ferrimagnets have a nonzero spontaneous magnetization, but not necessarily aligned local moments.

Detailed information is on the link below:

https://services.math.duke.edu/~ng/math/papers/senior-thesis.pdf

Ising Model

 The Ising model is the prototype model for all magnetic phase transitions and is probably the most studied model of statistical physics. In this model, the spin operator b S~ is replaced by a number, which represents the z− component of the spin and is usually S = ±1 (”up” or ”down”).

More information is on the link below:

http://www-f1.ijs.si/~vilfan/SM/ln2.pdf