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Physical review. B, Condensed matter

Exact differential renormalization of the Ising model at criticality

H. J. F. Knops, H. J. Hilhorst

Published Apr 1, 1979• 13 citations
Publisher

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.

DOI
10.1103/physrevb.19.3689

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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.
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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 $\ensuremath{\alpha}=0$, 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.
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