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The magnetic properties of the mixed spin-1/2 and spin-1 Ising ferrimagnetic system with nearest-neighbor and crystal-field interaction on the Bethe lattice are investigated within the framework of exact recursion relations. Particular interest is given to the coordination numbers q = 3 , 4, 5 and 6. The existence and dependence of compensation temperature on crystal-field interaction and coordination number is also investigated. If q ⩾ 4 , we find that the compensation temperature point is obtained in a restricted region of crystal-field interaction Δ .

We formulate the Ising model with dipole–quadrupole interaction on a Cayley tree. As a function of coordination number, we discuss the reentrant character of phase boundaries. A critical line separates the disordered and antiferromagnetic phases. An interesting fact is that the model does not have any tricritical and multicritical points.

The ferromagnetic Ising model on a Cayley tree of coordination z ( z=3 , 4 and 5), is investigated through the damage-spreading technique, within the heat-bath dynamics. Two distinct regimes are found, depending on the temperature: a low-temperature one, for which the initial damage may propagate, and a high-temperature one, for which it does not. These two regimes are separated by a damage-spreading temperature, T d (z) , which seems to coincide with the Curie temperature of the corresponding Bethe lattice, T c (z) . It is shown that as the temperature approaches T d (z) from below, the long-time damage goes to zero smoothly, contrary to the sharp behavior observed for ferromagnets in regular lattices. For T d (z) , the long-time damage shows a dependence on the initial conditions (magnitude and allocation of the initial damage), as well as on the particular choice of the sequence for updating the spins.

The q-state Potts model is studied on a Bethe lattice in which each spin interacts with n nearest neighbors through one kind of interaction and with the other nearest neighbors through interactions of another kind. The emphasis is put on the determination of the influence of parameters like n, the branching ratio z and the number of states q on the phase diagram.

The mean-field approximation of the one-dimensional ferromagnetic p -state Potts model with interactions decaying as a power law is developed using a modified Bethe lattice recently introduced. It is shown that the transition is of second order (first order) for p ⩽ 2 ( p > 2). The critical behavior is also presented.

Lattice spin systems on Bethe lattices or Husimi trees can be studied recursively. We present a new criterion for the location of phase transitions for such systems. We then apply the criterion to three different systems with complex phase diagrams, in particular with transitions at non-zero magnetic fields resulting from multi-site interactions.

The antiferromagnetic Potts model in the magnetic field is rigorously considered on the Bethe lattice by means of a recursion relation. This allows one to study the critical properties of the model as the properties of the iteration sequence { x n } in the limit n →∞. We observe the line of the second order phase transition, occuring as a period doubling bifurcation of the recursive sequence.

A Green function formalism is employed to study the effect of nonuniaxial single-ion anisotropy on the magnetic excitations in a dilute ferromagnet. The local density of spin wace states (LDOS) is obtained via diagonalizing the transfer matrix in the Bethe lattice approximation. The dependence of LDOS and the average magnetization on the anisotropy strength is calculated numerically.

The magnetic properties of a diluted Heisenberg ferromagnet in which the lattice sites are occupied randomly by magnetic and non-magnetic atoms are studied using a self-consistent cluster-Bethe-lattice method. The magnetization, Curie temperature and susceptibility are calculated for different concentrations of the magnetic atom, and are compared with the results of the mean-field approximation.

An extension or modification of the usual Bethe lattice is used to approximate one-dimensional Ising models with long-range pair interactions of the form Σ ( J / m Θ )σ i σ i + m . In particular an expression is obtained which allows one to one to obtain the critical temperature, T c , as a function of Θ, in the approximation. Our results are compared to a number of other T c approximations.

The spectrum of magnetic excitations at a disordered interface between two ferromagnetic materials was studied by using the Bethe lattice model and the Green function method. The effect of the interface composition and chemical short-range order on the local density of states (LDOS) is examined.

Within a Bethe lattice approach we obtain the first-neighbor, temperature dependent, effective pair potential for the hole-induced frustration Ising model, in the short-range potential approximation. Frustration- and fluctuation-driven pairing tendencies occur at distinct ranges of the temperature scale for low hole concentrations. However, as concentration increases these two mechanisms overlap.

We show that the free energy functional of the p-state Potts model in the Bethe-Peierls approximation is mapped by the combined operations of dilation and inversion of the magnetic field and of the order parameter, on the free energy of the p′(= p (p−1) ) -state model. The study of this functional in the interval pϵ (1, 2) is sufficient to determine completely the thermodynamic behavior of the system for all p > 2.

A theoretical investigation of the effects of the site dilution on spin waves in a transverse Ising model is presented. The local density of spin-wave states is calculated within the framework of the Green function theory by the application of the cluster Bethe lattice method. Both ferromagnetic and paramagnetic phases are considered. In the paramagnetic phase, a decrease of the concentration p of spins will lead to the appearance of localized states whose contribution to the local density of states keeps increasing. In the ferromagnetic phase, the existence of spontaneous magnetization, because of its dependence on the concentration of the magnetic atoms, gives rise to a quite different behavior of the local density of states. Some other interesting results are also presented.

We use the analogue of the cluster theory of Ursell and Mayer1 for a ferromagnetic Ising model in the presence of a magnetic field to describe the Ising singularity. In this cluster theory, the Ising model is described by a set of diagrams, which resemble branched polymers, characterized by the total number of bonds B and the total number of odd vertices V.

Calculations of the density of states of the Anderson model Hamiltonian are reported for the case of purely off-diagonal disorder for a Bethe lattice. With increasing off-diagonal disorder, the density of states exhibits a band-center peak at E=0, where the ordered lattice did not have a Van Hove singularity. In the light of results of other theories, the present data suggest strongly the universality of this feature in respect of the dimension and the structure of the lattice.

A tricritical point introducing dipolar and quadropolar exchange interaction in a spin-1 Ising model is defined exactly. The phase diagrams for different values of the dipolar and quadropolar exchange interactions are given and the second-order phase transition defined by the quadropolar moment is found. The tricritical exponents are calculated exactly.

Exact analytic formulae are obtained for the electronic structure of an ordered binary alloy in a Bethe lattice described by a tight-binding hamiltonian that includes nearest neighbour and second nearest neighbour interactions. Some interesting results are obtained and discussed in the text.

We present here a new treatment of short-range order in a disordered binary alloy with Bethe lattice topology based on the single-site approximation. We show that our method yields the limits of segregation and binary sequence exactly, while it reduces to the CPA in the case of random order.

By modifying the mixed-Bethe-lattice model, the properties of spin wave spectra of a ferromagnetic supperlattice are studied. The local density of magnon states (LDOMS) are obtained.

A new theory for incorporating short range order (SRO) in the cluster Bethe lattice method (CBLM) for disordered binary alloys is proposed. It is applied to determine the condition for the appearance of moments in the disordered phase of a narrow-band metal for general values of concentration and SRO parameter. The final result (i) yields all the known limiting conditions exactly and (ii) displays in a transparant form the SRO dependence, indicating the merit of the present approach.

The Green function of a particle that performs a random-walk on the Bethe lattice in the presence of the Coulomb potential is obtained in terms of the incomplete beta function. The associated continuous local density of states exhibits a Lifshitz tail. The discrete eigenvalues are also obtained analytically.

The exact expression for the Curie temperature of a spin-1 Bethe lattice with dipolar and quadrupolar interactions is derived and the nature of variation of the Curie temperature with quadrupolar interaction constant is studied.

The two-spin correlation function of the Ising model on a Bethe lattice as a function of temperature and magnetic field. It is found that the coherence length is finite at finite temperatures.

The size of the critical region for long- but not infinite-range Ising spin glasses is calculated by looking at the single loop corrections to the Bethe lattice approximation. The size of the critical region is predicted to go as 1 Z for large Z, where Z is the number of interacting neighbors.

For d=1, percolation clusters follow a scaling law with critical exponents σ=1 and τ=2. For the limit d→1, critical exponents can differ from their d=1 values, a complication which can already be studied in the simple Bethe lattice solution for cluster numbers.

Description is given of the Bethe lattice and the Weiss lattice on Reimann sheets. It is shown that the spin pair correlation function of the Ising model on these lattices, is directly related with the corresponding quantity in the Bethe and the Weiss approximation, respectively.