# Internal energy of the 2D Ising model

The above domain coloring plot shows an analytic continuation to the complex plane of the internal energy of the 2D Ising model. The domain variable is $z=\tanh J \beta$, which is a standard high temperature variable used to study the 2D Ising model.

I originally made the plot in 2014 soon after I first began to study the exact solution of the Ising model. A few days ago, I happened to find this old picture from 2014 and decided to share it here.

What a beautiful model! I had forgotten just how beautiful the structure is.

The 2 circles represent thermodynamic limit of the zeroes of the partition function. In this limit, singularities appear where the zeroes would be, as can be seen from Onsager’s formula. The critical temperature of the Ising model corresponds to the singularities on the real line, of which there are 4. The two singularities closer to the origin on the real line give the well known positive and negative critical temperatures $\pm \beta_c J \approx 0.44$. The other two give physically non-realizable complex critical temperatures, which are related to the standard critical temperatures via a symmetry known as the inversion relation (see MT Jaekel and JM Maillard, Symmetry relations in exactly soluble models, Journal of Physics A: Mathematical and General, 1982).

Mathematica Code:

f[z_] := (-\[Pi] (1 + z^2)^2 +
2 (1 – 6 z^2 + z^4) EllipticK[(16 z^2 (-1 + z^2)^2)/(1 + z^2)^4])/(
2 \[Pi] (z + z^3));

paint[z_] :=
Module[{x = Re[z], y = Im[z]},
color = Hue[Rescale[ArcTan[-x, -y], {-Pi, Pi}]];
shade = Mod[Log[2, Abs[x + I y]], 1];