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.

Physical Review LettersJun 19, 197838 citations

Physica A Statistical Mechanics and its Applications

Exact differential renormalization group equations for Ising models on square lattices

Wim van Saarloos

Published May 1, 1982• 1 citation

Publisher

DOI: 10.1016/0378-4371(82)90209-6

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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.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.

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Physical Review LettersJun 19, 197838 citations

{y}_{T}=1

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