Some of the properties of the bond diluted s-state Potts model are examined in this paper, and in particular the phase boundary It is shown that as s to 1, the model is simply related to the bond percolation problem By extending the replica arguments of Domany (1978) (which are probably exact), it is shown that near p, the phase boundary behaves as exp(-sK)=(ln s)/(s-1)(p-pc/pc) for all 2-D lattices The authors also construct a coherent potential approximation that is exact for all lattices as s to 1, and for arbitrary s on Bethe lattices

The dilute random-bond Potts model

The thermodynamics of the site randomly diluted Ising model calculated in the quenched limit are studied. It is shown that in general the partition function of the quenched site randomly diluted Ising model can be obtained by a simple transformation from the partition function of the quenched site randomly diluted Ising model defined with decorated bonds between its random sites. This transformation is very useful as the decorated random systems yield easily and quite successfully to approximate methods. An approximation for the energy and specific heat of the quenched diluted Ising model on the honeycomb lattice is obtained by using this transformation in conjunction with an approximation for the partition function of the quenched decorated random system.

Thermodynamics of the randomly diluted Ising model on the honeycomb lattice

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-1/2 Ising 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.

Critical frontiers associated with the bond-diluted Ising ferromagnet on triangular and honeycomb lattices

The bond disordered two-dimensional Ising model is studied for a small concentration of randomly substituted defects. The fermion mapping is used and the development is in terms of alloy scattering theory. Three approximations are used. At the lowest level, the ATA reproduces the results of the superlattice model. The CPA is equivalent to the annealed model. An adequate treatment of the quenched model requires the leading contributions from all cross-line diagrams to be included. This is accomplished by parquet theory, and the scattering from three channels is evaluated. Cancellation between two of the channels leaves the third channel as the only contributor. Three conflicting estimates of the specific heat behaviour of the quenched model have appeared in the literature recently. The results of this paper agree with those of Dotsenko and Dotsenko (1982), namely ln(-ln mod T-Tc mod ).

Critical specific heat of 2D Ising model in presence of bond defects

A systematic cluster extension of the effective-interaction approximation for quenched-bond disorder in random Ising systems is proposed. For bond dilution, exact results for the critical concentrations are obtained in the single-bond approximation for the self-dual square lattice and in the three-bonds approximation for the dual triangular and honeycomb lattices. Qualitative changes in the critical line are observed in the frustration model when going from the single-bond to the three-bonds approximations.

Cluster extension of the effective-interaction approximation. I. Quenched-bond disorder in the Ising model

The density matrix of an Ising model with quenched random interactions is written in terms of a translationally invariant effective Hamiltonian, Heff, with a random perturbation Heff is determined by requiring, as in the coherent potential approximation, that repeated scatterings from a single site do not affect the partition function It is shown that Heff describes an annealed random system

A B Harris

Papers

The coherent potential approximation for the density matrix of a random alloy