Exact differential renormalization of the Ising model at criticality

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Abstract
We analyze the exact real-space renormalization transformation recently discovered for the Ising model. The transformation, which is a set of differential equations for three spatially varying nearest-neighbor interactions, is specialized to the subspace of critical systems. First the exact solution is presented to the eigenvalue problem for the linearized flow around the previously reported fixed point. We find that the stability properties of this fixed point can only be obtained from the nonlinear equations. By exhibiting a special class of solutions to the nonlinear equations we show that the fixed point is unstable. Among the solutions we find three lines of fixed points emerging from the original fixed point.