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Journal of Physics C Solid State Physics

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

Deepak Kumar

Published Aug 14, 1979• 1 citation
Publisher

Abstract

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.

DOI
10.1088/0022-3719/12/15/003

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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.
The bond-diluted classical Heisenberg ferromagnet is treated by a position space renormalisation group method valid when the valid behaviour is at low temperatures. The method is outlined and used to obtain exact (one-dimensional) and approximate (two- and three-dimensional) results for the critical behaviour of the correlation length near the percolation limit. The procedure agrees in the pure limit with exact results of Polyakov and Migdal for the two-dimensional case. At general concentrations an exact correspondence with related treatments of random resistor networks is exhibited. This implies that the percolation-to-thermal crossover exponents of the Ising and Heisenberg systems are the same. The generalisations required for a proper comparison with neutron scattering experiments are briefly discussed.
Treats a percolation model with anisotropic bond occupation probabilities in order to study the crossover behaviour of the effective spatial dimension of the system. Previous position-space renormalisation group (PSRG) studies of this problem on the square lattice show a crossover from two-dimensional to pseudo-one-dimensional critical behaviour. To investigate the possible reasons for this surprising result, which is the opposite of the behaviour observed in thermal critical phenomena, the authors first develop alternative PSRG schemes. They study the predictions of these groups for asymptotically large rescaling parameters where the approximations in the PSRG are thought to become negligible. From a consideration of the approximations involved, they are led to a decimation transformation that uses an anisotropic cluster. The results of this method indicate that anisotropic percolation is in the same universality class as isotropic percolation, in complete analogy with thermal critical phenomena.
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.
By using the replica trick and a duality transformation, the bond-diluted random Ising model is mapped onto a new Hamiltonian. This demonstrates the higher order critical nature of the percolation point and identifies the appropriate crossover scaling variables. Taking the n to 0 limit of the replica method near the percolation point is shown to be equivalent to the q to 1 limit of the Potts model. The critical line near p c is calculated, yielding for a square lattice exp(-2K c )=2ln2(p- 1 / 2 )+0(p- 1 / 2 ) 2 .