Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Physics C Solid State Physics

Renormalisation group approach for percolation-magnetic indices

L.G. Marland, Robin B. Stinchcombe

Published Jun 28, 1977• 22 citations
Publisher

Abstract

A renormalisation group approach developed for the percolation problem by Young and Stinchcombe (1975) is extended to yield magnetic indices. A real-space cluster calculation is carried out for the Ashkin-Teller-Potts model (Ashkin & Teller 1964, Potts 1952) with general S, allowing contact with results for the Ising model (S to 2) as well as with the previously published thermal indices of the percolation problem (S to 1). For the percolation problem, the magnetic index obtained is 2.06, about 10% larger than current series estimates, and results in gamma =2.87.

DOI
10.1088/0022-3719/10/12/026

Related papers

A real-space renormalisation group for the triangular lattice

N -C Chao

The author extends the real-space renormalisation group approach of Reynolds, Klein and Stanley (1977) to the triangular lattice using a three-bond cluster as an elementary cluster. His results for the bond-percolation and Ising fixed points show considerable improvement over those of the decimation results of Yeomans and Stinchcombe (1979) and agree with the exact values to within 11% error.

Journal of Physics C Solid State Physics•Jun 20, 1981•4 citations

A renormalisation group investigation of the Ashkin-Teller-Potts model

S E Ashley

The Ashkin-Teller-Potts model is examined using two renormalisation group methods: (a) the cluster expansion; and (b) the lower bound variational method. The critical surface is found to be in agreement with that conjectured by Wu and Lin (1974). The authors find a very strong indication of a line of fixed points along which they obtain a continuous variation of the specific heat exponent with the four-spin coupling. A comparison is made with the eight-vertex model.

Journal of Physics A Mathematical and General•Oct 1, 1978•11 citations

Simple Method to Calculate Percolation, Ising, and Potts Clusters: Renormalization-Group Applications

Constantino Tsallis, Silvio V. F. Levy

We introduce a procedure which considerably simplifies the calculation of clusters like those commonly appearing in real-space renormalization-group treatments of bond-percolation and pure and random Ising and Potts problems. The method is illustrated through two applications for the $q$-state Potts ferromagnet.

Physical Review Letters•Sep 28, 1981•97 citations

Calculation of the percolation-thermal crossover index for the bond-diluted Ising model

Deepak Kumar

A position-space renormalisation group (RG) calculation, which uses the replica trick and the Migdal approximation, is reported. The use of the replica trick allows the percolation-thermal crossover index phi to be determined without the ambiguities that are usually encountered in real-space RG calculations. A value of phi =1 is found for all dimensions, in agreement with results obtained previously.

Journal of Physics C Solid State Physics•Aug 14, 1979•1 citation

New clusters for the real-space renormalisation group

N -C Chao

The author proposes a new type of cluster in addition to those of Reynolds, Klein and Stanley for the square lattice and another in addition to those of Chao for the triangular lattice for a real-space renormalisation group approach for studying critical phenomena. Thus each lattice is associated with two types of clusters-the longitudinal and the diagonal. The results for the bond percolation and Ising fixed points obtained from the diagonal square and honeycomb clusters agree with the exact values to within 6-11% error; whereas those obtained from the longitudinal triangular and honeycomb clusters agree with the exact values to within 2-4.5% error. The critical exponents corresponding to the percolation and the Ising problems also show reasonably good results: in particular those calculated by using the longitudinal triangular cluster agree with the exact values to within 6.7% and 3.2% error respectively.

Journal of Physics C Solid State Physics•Dec 20, 1981•3 citations

Critical properties of the Ashkin-Teller model from the mean-field renormalisation group approach

J. A. Plascak, F. C. Sá Barreto

By using small size clusters, which carry all the basic symmetries of the Ashkin-Teller model, the authors obtain within the mean-field renormalisation group approach the complete phase diagram of the model. Estimates of the thermal critical exponent are also obtained.

Journal of Physics A Mathematical and General•Aug 1, 1986•35 citations
A renormalisation group approach developed for the percolation problem by Young and Stinchcombe (1975) is extended to yield magnetic indices. A real-space cluster calculation is carried out for the Ashkin-Teller-Potts model (Ashkin & Teller 1964, Potts 1952) with general S, allowing contact with results for the Ising model (S to 2) as well as with the previously published thermal indices of the percolation problem (S to 1). For the percolation problem, the magnetic index obtained is 2.06, about 10% larger than current series estimates, and results in gamma =2.87.
We introduce a procedure which considerably simplifies the calculation of clusters like those commonly appearing in real-space renormalization-group treatments of bond-percolation and pure and random Ising and Potts problems. The method is illustrated through two applications for the qqq-state Potts ferromagnet.
The author proposes a new type of cluster in addition to those of Reynolds, Klein and Stanley for the square lattice and another in addition to those of Chao for the triangular lattice for a real-space renormalisation group approach for studying critical phenomena. Thus each lattice is associated with two types of clusters-the longitudinal and the diagonal. The results for the bond percolation and Ising fixed points obtained from the diagonal square and honeycomb clusters agree with the exact values to within 6-11% error; whereas those obtained from the longitudinal triangular and honeycomb clusters agree with the exact values to within 2-4.5% error. The critical exponents corresponding to the percolation and the Ising problems also show reasonably good results: in particular those calculated by using the longitudinal triangular cluster agree with the exact values to within 6.7% and 3.2% error respectively.