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Journal of Physics C Solid State Physics

Renormalisation group approach for the two dimensional random Ising model

Gary S. Grest

Published Apr 14, 1978• 10 citations
Publisher

Abstract

The Niemeyer-van Leeuwen formulation of the renormalisation group is used to describe a quenched dilute Ising model on a triangular lattice in two dimensions. The two- and three-cluster approximations are applied to calculate the probability dependence of the critical temperature T c (p) and the critical exponents near the percolation threshold phi =1 is found to be in agreement with previous epsilon expansion results.

DOI
10.1088/0022-3719/11/7/022

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The Niemeyer-van Leeuwen formulation of the renormalisation group is used to describe a quenched dilute Ising model on a triangular lattice in two dimensions. The two- and three-cluster approximations are applied to calculate the probability dependence of the critical temperature T c (p) and the critical exponents near the percolation threshold phi =1 is found to be in agreement with previous epsilon expansion results.
The Niemeijer-van Leeuwen formulation of the renormalisation group has been used to discuss the quenched dilute Ising model and the Ising spin-glass in two dimensions. The authors chose a five-spin cell which, in the pure case, treats the Ising antiferromagnet equally as well as the ferromagnet. This choice of cell leads to the existence of a spin-glass ground state which was found to be stable for the whole composition range. The spin-glass transition was found to be sharp and the corresponding critical exponents were calculated. For the dilute Ising model the behaviour of the reduced critical temperature T c (p)/T c (1), the percolation concentration and the correlation length exponent were found to be in good agreement with the exact results and series calculations.
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.
A renormalization-group technique is used to study the critical behavior of spin models in which each interaction has a small independent random width about its average value. The cluster approximation of Niemeyer and Van Leeuwen indicates that the two-dimensional Ising model has the same critical behavior as the homogeneous system. The $\ensuremath{\epsilon}$ expansion for nnn-component continuous spins shows that this behavior holds to first order in $\ensuremath{\epsilon}$ for n>4n>4n>4. For n<4n<4n<4, there is a new stable fixed point with $2\ensuremath{\nu}=1+[\frac{3n}{16(n\ensuremath{-}1)}]\ensuremath{\epsilon}$.
A two-step renormalisation group approach-a Migdal-Kadanoff (or decimation) transformation (MK) followed by a mean-field renormalisation group (MFRG)-has been applied to random bond Ising models on d-dimensional hypercubic lattices. Calculations of the critical couplings and percolation concentrations for two and three dimensions show much improvement, even for small size clusters, compared with the results of separate MK or MFRG calculations. Critical exponents nu T and nu p are also improved in some cases. Estimates of critical curves in the concentration-temperature plane are also obtained for the two- and three-dimensional models.
The equivalence of the critical behavior of the ddd-dimensional Ising model at a finite temperature to that of a ddd- 1 dimensional system at zero temperature as a function of a transverse magnetic field is used to calculate the critical exponents for the three-dimensional model by applying the Niemeijer-Van Leeuwen renormalization-group method to the two-dimensional problem.