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Physical Review Letters

Differential Form of Real-Space Renormalization: Exact Results for Two-Dimensional Ising Models

H. J. Hilhorst, Michael Schick C, J. M. J. van Leeuwen

Published Jun 19, 1978• 38 citations
Publisher

Abstract

A new real-space renormalization method is developed which leads to differential renormalization-group equations. For a two-dimensional triangular Ising lattice these equations can be solved exactly and yield and exact critical properties.
DOI
10.1103/physrevlett.40.1605

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Physical Review Letters•Apr 21, 1980•13 citations
A previously developed real-space renormalization-group approach to critical dynamics is examined. A correction to the formalism has to be introduced. The need for this correction is demonstrated through an exactly soluble one-dimensional Ising model. The new version imposes additional serious limitations on the usefulness of the method within standard approximation schemes. The second-order cumulant approximation of the two-dimensional kinetic Ising model on a triangular lattice is used to demonstrate these limitations.
We present an exact differential renormalization-group method for the square Ising model. This method is analogous to that introduced by Hilhorst et al. for the triangular Ising model. Our approach is based upon the construction of such a renormalization transformation which remains confined to the space of four spatially dependent nearest-neighbor interactions. We derive the renormalization-group equations in the form of a set of four linear first-order partial differential equations for the interactions. We obtain a class of solutions of these equations and determine a nontrivial fixed point. The analysis of the flow around the fixed point shows that this fixed point is stable with respect to perturbations within the critical surface and is unstable in the temperaturelike direction. A unique eigenvalue, yT=1{y}_{T}=1yT​=1, from the eigenvalue equation for the temperature direction is found, in accordance with the exact Onsager solution.
The two- and three-dimensional transverse-field Ising models with ferromagnetic exchange interactions are analyzed by means of the real-space renormalization-group method. The basic strategy is a generalization of a method developed for the one-dimensional case, which exploits the exact invariance of the model under renormalization and is known to give the exact values of the critical point and critical exponent ν. The resulting values of the critical exponent ν in two and three dimensions are in good agreement with those for the classical Ising model in three and four dimensions. To the best of our knowledge, this is the first example in which a real-space renormalization group on (2+1)- and (3+1)-dimensional Bravais lattices yields accurate estimates of the critical exponents.
The real-space renormalization-group method for the dynamics of critical phenomena introduced in the previous paper is applied to two calculations within the two-dimensional kinetic Ising model on a triangular lattice. In both cases a cumulant expansion in the intercell couplings is performed. In the first calculation are found the most general first-order recursion relations coupling neighboring cells. From these recursion relations is extracted the dynamic critical exponent z=1.70z=1.70z=1.70. While this is a reasonable first-order result, the results for the statics are not as satisfactory. In the second calculation it is shown how the terms that are generated at second order are important in determining the static results. We obtain at second order z=2.22z=2.22z=2.22, together with a critical coupling Kc=0.251{K}_{c}=0.251Kc​=0.251 and a thermal critical index $\ensuremath{\nu}=0.950$.
When differential real-space renormalization-grup theory was proposed by Hilhorst, Schick, and van Leeuwen, they suggested that their approach could only be applied to lattice models for which a star-triangle transformation exists. However, differential renormalization-group equations for the square Ising model have recently been proposed whose derivation does not involve the star-triangle transformation. We show that the latter equations are not exact renormalization-group equations by an analysis that reveals some essential limitations of the present formulation of differential real-space renormalization. We investigate the structure of the renormalization-group flow equations obtained in this method and uncover a strong property of these equations that simplifies the calculations in actual applications of the theory. However, the status and implications of this property, which embodies the crux of the theory, are not yet fully understood.