Bethe lattice approximation of long-range interaction Ising models

James L. Monroe

The mean-field approximation and the Bethe approximation are two of the most often used approximations when one wants to obtain approximations of the phase diagrams and the critical temperature of lattice spin systems. Both can and have often been generalized to produce what are known as cluster mean-field and Bethe approximations. Generally, three characteristics are associated with these approximations. First, they give upper bounds to the critical temperature; second, considering larger clusters will result in better approximations; and third, the Bethe approximation is better than the corresponding mean-field approximation. We show what we believe to be a rather surprising result that, for one-dimensional Ising models with algebraically decaying interactions falling off slowly enough, the Bethe cluster approximations violate all three of these characteristics.

Ladislav Šamaj

Sze-Shiang Feng, Qiu Xijun

The critical exponents of the triangular lattice Ising model with long-range interactions are calculated by the real space renormalization group. Using the simplest Kadanoff blocks and the lowest approximation of cumulant expansion, it is shown that there exists a finite critical temperature when .

R. Z. Bariev, M.P. Zhelifonov

Copenhagen O Dk

The partition function of the 2D Ising model with random nearest-neighbour coupling is expressed in the dual lattice made of square plaquettes. The dual model is solved in the mean field and in different types of Bethe-Peierls approximations, using the replica method. The application of methods of mean-field type to Ising models allows one to obtain very accurate approximations of the thermodynamic quantities. However, in the presence of quenched disorder this approach is difficult to implement. In this paper, we show that rather good results can be obtained after performing a duality transformation of the Ising model with random nearest-neighbour coupling that assumes the values Jij D1 with equal probability. The model is thus defined on a dual lattice where the spin variables are attached to the square plaquettes. The advantage is that the quadratic term of the dual Hamiltonian has constant coefficients instead of random ones. It is therefore possible to use the standard methods of the mean field to estimate the quenched free energy. Our results can be generalized to higher dimensions, although the approximations become rougher, because the number of plaquette spins over the number of interaction links increases with the dimensionality (1). In particular, we obtain an extremely good estimate of the ground- state energy of the random Ising model, by applying the Bethe-Peierls approximation where part of the short-range order is taken into account. In section 1, we introduce the dual lattice made of elementary square plaquettes. On this lattice the partition function can be expressed as a function of the inverse temperature e D 1 ln tanh.1=T/ whereT D 1 is the temperature of the original lattice. In section 2, we apply the mean-field approximation to the dual model, using the replica method. In section 3, we introduce the Bethe-Peierls approximation. This allows us to obtain a very precise estimate of the ground-state energy of the two-dimensional Ising model with random coupling. In section 4, we show that it is possible to improve the Bethe-Peierls approximation by introducing an interaction between different replicas. In section 5, the reader will find some remarks and conclusions.

S. H. Florez, M. Roldán