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.

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