The Ising Model

Abstract

There are very few models of interacting spins which are exactly solvable by the methods of statistical physics. Among these, the Ising model is exactly solvable in one dimension [128]. In this case, fluctuations are so large that the phase transition is pushed right down to absolute zero temperature. In two dimensions, the model is still exactly solvable in zero field, a result due to Onsager (1944) [129, 130]. Phase transition now occurs at a finite critical temperature. Depending on the magnetic coordination z of the lattice, the critical temperature k b T c is 1.51866 × J for the honeycomb lattice (z = 3), 2.26918 × J for the square lattice (z —4), and 3.64096 × J for the triangular lattice (z = 6). Not only does the critical temperature differ from that given by mean field theory, but critical exponents (α = 0, β = 1/8, γ = 7/4, δ = 15 and v = 1) do not coincide with those in Landau theory. Onsager’s solution gives a concrete example of the significance of fluctuations. These strongly renormalise critical behaviour. Hence the importance of the 2-dimensional Ising model as a test for modern theories of phase transition.