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Physical review. B, Condensed matter

Use of the star-triangle transformation for the application of differential real-space renormalization-group theory

Wim van Saarloos

Published May 1, 1983• 1 citation

Abstract

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.

DOI: 10.1103/physrevb.27.5678

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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.

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,

{y}_{T}=1

, from the eigenvalue equation for the temperature direction is found, in accordance with the exact Onsager solution.

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.

A real-space renormalization-group transformation is constructed between blocks of linear dimensions

\mathrm{bL}

and

L

to study the ground-state properties of quantum lattice models. In this scheme it is possible (i) to estimate the errors due to the irrelevant variables, including boundary effects, and (ii) to test scaling explicitly, so that an accurate calculation can be performed over the entire range of the parameter space. Application to the one-dimensional Ising model in a tranverse field yields excellent results.

A general method for optimizing real-space renormalization-group transformations is presented. It is applicable to both spin systems and lattice gauge theories, with use of Monte Carlo computer simulations. It provides substantial improvement in the convergence of critical exponents for the

d=3

Ising model.