The Ising Model

Also shown in the table are results for the Ising model. The Ising [15]model is a very simple model of ferromagnetic systems. The model in $d$-dimensions consists of a periodic lattice (hexagonal, cubic etc) with fixed lattice points. Attached to each lattice site $i$ are ‘spin variables’ $s_i$ which take either the value $+1$ or $–1$. These spin variables represent the ‘atomic magnets’. Each spin is allowed to interact with only its nearest neighbours or an external magnetic field. The spin-spin interaction is such that it is energetically favourable for neighbouring spins to align. Therefore at low temperatures, where thermal fluctuations are small, one might expect that the thermodynamically stable state will correspond to a state of spontaneous magnetization (even in the absence of an external magnetic field). That is, although the spins only interact with their neighbours, the net result can be a cooperative state, in which far-away spins become correlated. Thus one sees the emergence of long ranged (macroscopic) correlations and order at low enough temperatures even though the microscopic model has only short-ranged interactions. Quantitatively, it is remarkable that such a crude model gives theoretical predictions for the critical exponents and scaling relations that agree with experiments on real systems. This again illustrates the power of universality near the critical point: the microscopic (short-distance) information of the system, whether real or theoretical, is ‘washed out’, leading to common and similar macroscopic properties for systems within each universality class. (Universality classes differentiate between systems in different dimensions with different underlying symmetries). Although the Ising model is simple to state, the computation of its thermodynamic properties is extremely involved and for $d=3$ requires numerical effort using a computer. Nevertheless many other models, as ‘simple’ as the Ising model but different from it have been used to model real systems and to compute and compare their critical properties. This is not only a test of universality but gives insight into properties of varied systems.