Spin systems on hypercubic Bethe lattices: A Bethe-Peierls approach

The inverse Ising problem consists in inferring the coupling constants of an Ising model given the correlation matrix. The fastest methods for solving this problem are based on mean-field approximations, but which one performs better in the general case is still not completely clear. In the first part of this work, I summarize the formulas for several mean- field approximations and I derive new analytical expressions for the Bethe approximation, which allow to solve the inver...

We use an m-vicinity method to examine Ising models on hypercube lattices of high dimensions d>=3. This method is applicable for both short-range and long-range interactions. We introduce a small parameter, which determines whether the method can be used when calculating the free energy. When we account for interaction with the nearest neighbors only, the value of this parameter depends on the dimension of the lattice d. We obtain an expression for the critical temperature in...

Using an iteration technique, we obtain exact expressions for the free energy and the magnetization of an Ising model on a two - layer Bethe lattice with intralayer coupling constants J1 and J2 for the first and the second layer, respectively, and interlayer coupling constant J3 between the two layers; the Ising spins also couple with external magnetic fields, which are different in the two layers. We obtain exact phase diagrams for the system.

We present an exact treatment of the hysteresis behavior of the zero-temperature random-field Ising model on a Bethe lattice when it is driven by an external field and evolved according to a 2-spin-flip dynamics. We focus on lattice connectivities z=2 (the one-dimensional chain) and z=3. For the latter case, we demonstrate the existence of an out-of-equilibrium phase transition, in contrast with the situation found with the standard 1-spin-flip dynamics. We discuss the influe...

We exactly solve the ferromagnetic spin-1/2 Ising model on the Bethe lattice in the presence of an external magnetic field by means of the equations of motion method within the Green's function formalism. In particular, such an approach is applied to an isomorphic model of localized Fermi particles interacting via an intersite Coulomb interaction. A complete set of eigenoperators is found together with the corresponding eigenvalues. The Green's functions and the correlation f...

In this work we present a new method to calculate the classical properties of magnetic nanoparticles. Based on the Bethe-Peierls (pair) approximation, we developed a simple system of equations for the classical magnetization of spins at any position within the nanoparticle. The nearest neightbor pair correlations are treated exactly for Ising spins, and the method can be generalized for various lattice symmetries. The master equation is solved for the Glauber dynamics (single...

The phase diagram of the random field Ising model on the Bethe lattice with a symmetric dichotomous random field is closely investigated with respect to the transition between the ferromagnetic and paramagnetic regime. Refining arguments of Bleher, Ruiz and Zagrebnov [J. Stat. Phys. 93, 33 (1998)] an exact upper bound for the existence of a unique paramagnetic phase is found which considerably improves the earlier results. Several numerical estimates of transition lines betwe...

We consider the analytic solution of the zero temperature hysteresis in the random field Ising model on a Bethe lattice of coordination number $z$, and study how it approaches the mean field solution in the limit z-> \infty. New analytical results concerning the energy of the system along the hysteresis loop and first order reversal curves (FORC diagrams) are also presented.

In statistical physics, one of the standard methods to study second order phase transitions is the renormalization group that usually leads to an expansion around the corresponding fully connected solution. Unfortunately, often in disordered models, some important finite dimensional second-order phase transitions are qualitatively different or absent in the corresponding fully connected model: in such cases the standard expansion fails. Recently, a new method, the $M$-layer o...

We consider the Ising spin glass for the arbitrary spin S with the short- ranged interaction using the Bethe- Peierls approximation previously formulated by Serva and Paladin for the same system but limited to S=1/2. Results obtained by us for arbitrary S are not a simple generalization of those for S=1/2. In this paper we mainly concentrate our studies on the calcutation of the citical temperature and the linear susceptibility in the paramagnetic phase as functions of the di...