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Journal of Physics C Solid State Physics

New clusters for the real-space renormalisation group

N -C Chao

Published Dec 20, 1981• 3 citations
Publisher

Abstract

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.

DOI
10.1088/0022-3719/14/35/001

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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.
The renormalisation group method of Kadanoff is applied to the spin- 1 / 2 Ising model in two dimensions using a transformation applicable to both the square and triangular lattices. Two types of cluster are studied for the triangular lattice.The first cluster yields a new fixed point but the results for the critical temperature and critical exponents are in poor agreement with the known exact values. The second cluster maps directly onto the fixed point found by Kadanoff for the square lattice yielding excellent values for both the critical exponents and the critical temperature.
Within a real space renormalisation group framework (12 different procedures, all of them using star-triangle and duality-type transformations) the authors calculate accurate approximations for the critical frontiers associated with the quenched bond-diluted first-neighbour, spin-1/2Ising ferromagnet on triangular and honeycomb lattices. All of them provide, in both pure bond percolation and pure Ising limits, the exact critical points and exact or almost exact derivatives in the p-t space (p is the bond-independent occupancy probability and t identical to tanh(J/kBT)). The authors best numerical proposals lead to the exact derivative in the pure percolation limit (p=pc) and, in what concerns the pure Ising limit (p=1) derivative, to a 0.15% error for the triangular lattice and to a 0.96% error for the honeycomb one; in the intermediate region (pc<p<1), where the exact critical frontiers are still unknown, the worst error in the variable t (for fixed p) is estimated to be less than 0.27% for the triangular lattice and 0.14% for the honeycomb one.
A renormalization group approach in real space is applied to the bond percolating square lattice in two dimension using two scaling transformations proposed by Reynolds et al. (Reynolds's method) and by Yuge (Yuge's method). The renormalization transformation R b ( p ), the eigenvalue λ( b ) and the correlation length critical exponent v ( b ) are calculated exactly by changing the scale of length b =4 for Reynolds's method and b=5/2b=5/\sqrt{2}b=5/2​ and 6/26/\sqrt{2}6/2​ for Yuge's method. The true critical exponent v is estimated v =1.341 and 1.337 by extrapolation, respectively. Discussions are made on the results.
A simple renormalisation-group approach based on self-dual clusters is proposed for the two-dimensional nearest-neighbour spin- 1 / 2 Ising model on the square lattice; it reproduces the exact critical point. The authors calculate the internal energy and the specific heat for vanishing external magnetic field, spontaneous magnetisation and the thermal (y T ) and magnetic (y H ) critical exponents. The results obtained from the first four smallest cluster sizes strongly suggest the convergence towards the exact values when the cluster size increases. Even for the smallest cluster, where the calculation is very simple, the results are quite accurate, particularly in the neighbourhood of the critical point.
A real-space renormalization group is developed which renormalizes probabilities directly in the percolation problem. An exact transformation is given in one dimension, and a cluster approach is presented for other lattices. The results are in excellent agreement with series calculations for the critical percolation concentration p c (both site and bond), and in good agreement for the correlation length exponent nu p . Additionally, in one dimension, a field-like variable is included and the remaining exponents are calculated.