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Journal of Physics A Mathematical and General

Cluster decimation derivation of exact percolation-thermal crossover exponent in dilute spin models

Robin B. Stinchcombe

Published Apr 21, 1983• 4 citations
Publisher

Abstract

It is shown that the decimation method of renormalisation group theory in a restricted parameter space, applied to any cluster, gives the exact unit value for the percolation-thermal crossover exponent for bond-diluted Ising, Potts and anisotropic Heisenberg models.
DOI
10.1088/0305-4470/16/6/023

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A position space renormalisation group technique (decimation) is used to treat site- and bond-diluted Ising ferromagnets on square and triangular lattices. In every case the renormalisation equations lead to two fixed points, one corresponding to the pure Ising system and one to percolation. Linearising around the fixed points, eigenvalues and hence the critical exponents nu T , nu P (correlation length), alpha (specific heat) and phi (crossover) are obtained. Numerical iteration of the recursion relations provides the critical curves. Values of critical temperatures, critical concentrations, exponents, and of limiting slopes of critical curves are in very satisfactory agreement with all known exact or series results. Points on the critical curve flow into the Ising fixed point under the transformation. The sign of the pure Ising exponent alpha is consistent with this flow for the site-diluted cases but not for the bond-diluted cases.
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 general theory is given for the quenched dilute sss-state Potts and nnn-vector models in any dimension ddd. It is shown that for $T\ensuremath{\rightarrow}0$ at the percolation threshold the Potts thermal exponent ${\ensuremath{\nu}}_{T}$ equals the percolation exponent ${\ensuremath{\nu}}_{p}$, implying a crossover exponent $\ensuremath{\varphi}=1$, for any sss and ddd. For the nnn-vector model (n>1n>1n>1), ${\ensuremath{\nu}}_{T}=\frac{{\ensuremath{\nu}}_{p}}{{\ensuremath{\zeta}}_{R}}$, where ${\ensuremath{\zeta}}_{R}$ is a resistivity critical exponent. Agreement with recent experiments for two-dimensional dilute Ising and Heisenberg systems is excellent.
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.