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Communications in Theoretical Physics

Exact Decimation Approach with Mean-Field Approximation for Randomly Diluted Ising Models

TANG KUN-FA, HU JIA-ZHEN

Published Jun 1, 1988• 0 citations
Publisher

Abstract

A new renormalization approach developed by authors previously is generalized to disordered systems. Calculations of the critical temperature, concentration, exponents, and of limiting slope of critical curve for the random bond-diluted Ising model on the square lattice, even in the lowest approximation, are in very satisfactory agreement with all known exact or series results.

DOI
10.1088/0253-6102/9/4/405

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A new renormalization approach developed by authors previously is generalized to disordered systems. Calculations of the critical temperature, concentration, exponents, and of limiting slope of critical curve for the random bond-diluted Ising model on the square lattice, even in the lowest approximation, are in very satisfactory agreement with all known exact or series results.
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The randomly bond-dilute two-dimensional nearest-neighbor Ising model on the square lattice is studied by renormalization-group methods based on the Migdal-Kadanoff approximate recursion relations. Calculations give both thermal and magnetic exponents associated with the percolative fixed point. Differential recursion relations yield a phase diagram which is in quantitative agreement with all known results. Curves for the specific heat, percolation probability, and magnetization are displayed. The critical region of the specific heat becomes unobservably narrow well above the percolation threshold pc{p}_{c}pc​. This provides a possible explanation for the apparent specific-heat rounding in certain experiments.
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.