Exact differential renormalization group equations for Ising models on square lattices

The differential real space renormalization theory of Hilhorst et al. is applied to Ising models on square lattices with nearest-neighbour interactions only. The renormalization flow equations for these two interaction parameters contain two auxiliary parameters; these parameters have to be determined by solving two additional equations, one partial differential equation and one ordinary equation. The necessity of introducing these additional parameters is explained by arguing that for the present formulation of differential real space renormalization theory in d dimensions, at least d + 1 parameters are required. The concept of local fixed point is introduced; this fixed point can be determined by solving algebraic equations. The linearized flow around it describes local properties of the system and is therefore related to the critical properties of homogeneous Ising systems. We study temperature-like perturbations around the local fixed point and find a unique eigenvalue yT = 1, in agreement with the known exact result.