While mean-field models can provide us with a lot of information (particularly on the thermodynamics) on the properties of ferromagnets with or without structural coupling effects, a microscopic model can give us more. As expected, the tradeoff is complexity. The Ising model is widely studied, and presents us with the "simplest" approach to interpreting the properties of a magnetic system.
No analytical solution to the field dependent 2D/3D Ising model has been reported, and so approximate monte-carlo methods are usually employed to obtain estimates of the magnetic and thermodynamic properties of Ising lattices. The Metropolis method is the most widely used approach, and results are widely available. Still, there are two limitations of this approach that make it difficult to study a field/temperature dependent compressible Ising ferromagnet:
1: For each individual field and temperature value, an independent, converged monte-carlo run must be performed.
2: A compressible lattice does not equilibrate without the use of an external pressure term.
I considered the possibility of overcoming these difficulties by developing a method that results in the full thermodynamic description of an Ising lattice, by estimating the microcanonical energy, magnetization and entropy (degeneracy) terms. This way the partition function can be estimated, and field/temperature effects are quickly calculated. For more details, see the following publication:
"Thermodynamics of the 2D Ising model from a random path sampling method", J. S. Amaral, J. N. Gonçalves and V. S. Amaral, accepted for publication IEEE Trans. Magn. (2014) [preprint available
here].
I'll show here some of the outputs that can be quickly obtained by the random path sampling (RPS) method, using as example the 4x4 2D simple square lattice.
For each possible magnetization value of a 4x4 lattice, there are only a reduced number of non-degenerate energy values, and each energy value will have its number of possible configurations (degeneracy). If the system is fully ordered (M=1), the energy/degeneracy values are simply
| Energy | Degeneracy |
| -32 | 1 |
If we flip only one spin, we now have:
| Energy | Degeneracy |
| -24 | 16 |
Flipping another spin, we now have two possible degenerate states.
| Energy | Degeneracy |
| -20 | 32 |
| -16 | 88 |
Things become quite more complicated if the system is in a disordered state. The highest entropy will be for M=0, and the energy/degeneracy values even for such a small system quickly become numerous:
| Energy | Degeneracy |
| -16 | 4.9763e+05 |
| -8 | 5.0525e+07 |
| -4 | 1.0498e+08 |
| 0 | 2.8554e+08 |
| 4 | 2.13678e+08 |
| 8 | 1.3814e+08 |
| 12 | 3.7955e+07 |
| 16 | 7.7926e+06 |
| 20 | 4.2071e+06 |
| 32 | 1.2483e+05 |
With the complete set of magnetization/energy/degeneracy values (the so-called Joint Density of States, the partition function can be calculated and so thermodynamic properties, such as average absolute magnetization, average energy and specific heat, magnetocaloric effect (isothermal magnetic entropy change), etc. are quickly calculated, together with their complete dependence on temperature and field:
It is also possible to establish the temperature and field dependent free energy, which allows the estimate of "non-average" M values, i. e. the magnetization values of the spin configurations that minimize the free energy. For the 4x4 lattice at zero field we obtain:
The discretization of possible magnetization values becomes less pronounced for larger systems, and smoother magnetization curves are obtained. Magnetization/energy/degeneracy tables of the 4x4 system can be downloaded here, in Matlab cell format. Data sets for other lattices types, including for Ising lattices with anisotropic interaction values, and 3D lattices up to 512 spins are available on request.