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Ukrainian Journal of Physics

Phenomenological Renormalization Group and Cluster Approximation

Oleg Borisenko, V. Chelnokov, Vladimir Kushnir

Published Nov 15, 2015• 0 citations
Publisher

Abstract

We propose a modification of the Nightingale renormalization group for lattice spin and gauge models by combining it with the cluster decimation approximation. Essential ingredients of our approach are: 1) exact calculation of the partition and correlation functions on a finite lattice strip; 2) preservation of the mass gap or the second moment correlation length, computed in the infinite strip length limit, on each decimation step. The method is applied to studying the general two- and three-dimensional Z(N) models. A perfect agreement with exact results (whenever available) is found. An extension of the method to models with a continuous symmetry is briefly discussed.

DOI
10.15407/ujpe59.07.0655

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We propose a modification of the Nightingale renormalization group for lattice spin and gauge models by combining it with the cluster decimation approximation. Essential ingredients of our approach are: 1) exact calculation of the partition and correlation functions on a finite lattice strip; 2) preservation of the mass gap or the second moment correlation length, computed in the infinite strip length limit, on each decimation step. The method is applied to studying the general two- and three-dimensional Z(N) models. A perfect agreement with exact results (whenever available) is found. An extension of the method to models with a continuous symmetry is briefly discussed.
We propose a modification of the Nightingale renormalization group for lattice spin and gauge models by combining it with the cluster decimation approximation.Essential ingredients of our approach are: 1) exact calculation of the partition and correlation function on a finite lattice strip; 2) preservation of the mass gap or the second moment correlation length, computed in the infinite strip length limit, on each decimation step.The method is applied for studying general two and three dimensional Z(N) models.A perfect agreement with exact results (whenever available) is found.An extension of the method to models with a continuous symmetry is briefly discussed.
Some renormalization group approaches have been proposed during the last few years which are close in spirit to the Nightingale phenomenological procedure. In essence, by exploiting the finite size scaling hypothesis, the approximate critical behavior of model on infinite lattice is obtained through the exact computation of some thermal quantities of the model on finite clusters. In this work some of these methods are reviewed, namely the mean field renormalization group, the effective field renormalization group and the finite size scaling renormalization group procedures. Although special emphasis is given to the mean field renormalization group (since it has been, up to now, much more used to study a wide variety of different systems) a discussion of their potentialities and interrelations to other methods is also presented.
A simple hybrid of real space renormalisation group theory and finite-lattice calculations is described. The method is based on a lattice-restructuring transformation, with the criterion that the free energies of the original and restructured lattices be equal. The authors approximate the free energy per spin of the infinite lattice by an exact evaluation of the partition function for a small cluster. As a result of this transformation, the coupling constant of the restructured lattice is found to be temperature dependent. The method is illustrated by studying the Ising model in two and three dimensions, and two-dimensional Potts models. The calculations involve little more than is required for a Migdal-Kadanoff transformation, yet, the critical properties are found to be significantly improved in comparison. Detailed comparisons are also made with the global thermodynamic properties calculated within this approximation (free energy, internal energy, specific heat, spontaneous magnetisation) and exact or best known results for the models. The simplicity of the method allows for the study of more complex lattice models, and several possible applications are discussed.
An analysis is made of various methods of phenomenological renormalization based on finite-dimensional scaling equations for inverse correlation lengths, the singular part of the free energy density, and their derivatives. The analysis is made using two-dimensional Ising and Potts lattices and the three-dimensional Ising model. Variants of equations for the phenomenological renormalization group are obtained which ensure more rapid convergence than the conventionally used Nightingale phenomenological renormalization scheme. An estimate is obtained for the critical finite-dimensional scaling amplitude of the internal energy in the three-dimensional Ising model. It is shown that the two-dimensional Ising and Potts models contain no finite-dimensional corrections to the internal energy so that the positions of the critical points for these models can be determined exactly from solutions for strips of finite width. It is also found that for the two-dimensional Ising model the scaling finite-dimensional equation for the derivative of the inverse correlation length with respect to temperature gives the exact value of the thermal critical index.
An analysis is made of various methods of phenomenological renormalization based on finite-size scaling equations for inverse correlation lengths, the singular part of the free energy density, and their derivatives. The analysis is made using two-dimensional Ising and Potts lattices and the three-dimensional Ising model. Variants of equations for the phenomenological renormalization group are obtained which ensure more rapid convergence than the conventionally used Nightingale phenomenological renormalization scheme. An estimate is obtained for the critical finite-size scaling amplitude of the internal energy in the threedimensional Ising model. It is shown that the two-dimensional Ising and Potts models contain no finite-size corrections to the internal energy so that the positions of the critical points for these models can be determined exactly from solutions for strips of finite width. It is also found that for the two-dimensional Ising model the scaling finite-size equation for the derivative of the inverse correlation length with respect to temperature gives the exact value of the thermal critical exponent. PACS: 05.10.Cc, 64.60.Ak, 05.50.+q, 75.10.Hk 1 1
We present a quantum cluster solver for the spin- S Heisenberg model on a two-dimensional lattice. The formalism is based on the real-space renormalization procedure and uses the lattice point group-theoretical analysis and non-Abelian SU (2) spin symmetry technique. The exact diagonalization procedure is used twice at each renormalization group step. The method is applied to the spin-half antiferromagnet on a square lattice, and a calculation of local observables is demonstrated. A symmetry-based truncation procedure is suggested and verified numerically.