Exact renormalization group equations for the two-dimensional Ising model

By explicit construction we show the existence of an exact renormalization group for triangular Ising models with only nearest-neighbor interactions. The recursion relations take the form of a set of three quasilinear first-order partial differential equations for the interactions. We determine a nontrivial fixed point and study the linearized flow around it. This yields the specific-heat exponent $\alpha =0\mathrm{}$, in agreement with the Onsager and Houtappel solutions, and demonstrates universality. The free energy is expressed as the trajectory integral of an explicitly given function of the interactions.