Evidence for a transcritical bifurcation in the 2D Ising model

We find that the 2D Ising model is at a transcritical bifurcation involving the exchange of stability between two fixed points, similar to the Gaussian and Wilson-Fisher fixed points in 4D. Using perturbative normal-form theory--a method from dynamical systems for analyzing bifurcations--we find the simplest flow equations for the 2D Ising model. From this we predict that the flows of the inverse specific heat undergo a transcritical bifurcation near $D=2$. This is consistent with the conformal bootstrap method, which hints at the existence of two fixed points for $D<2$. We bring Onsager's exact solution to its normal form, which has a logarithmic singularity due to a `resonance' between the temperature and free energy eigenvalues. More broadly, our work seems to imply that such resonances can be understood as bifurcations in measurable quantities.