Analytic properties of the Onsager solution of the ising model

The analytic properties of the Onsager solution of the two-dimensional Ising lattice in zero external magnetic field are investigated. The free energy above the critical temperature Tc can be regarded as an analytic function in the u plane, where u=exp(-4J/kBT), with singularities at u=0, ∞, and exp(±4J/kBTc). But its analytic continuation below the critical temperature does not describe a stable thermodynamic phase. Similar statements are true for the free energy below the critical temperature. The famous logarithmic singularity is briefly discussed. Finally, the knowledge of the analytic properties is shown, by elementary considerations, to explain the nonanalytic behavior at vanishing external field below the critical temperature-a result first enunciated by Yang and Lee for the free energy in the presence of an external magnetic field