Spin glass with two replicas on a Bethe lattice

Geometrical frustration and cluster spin glass with random graphs

We develop a based on a sparse random graph to account for the interplay between geometric frustration and disorder in cluster magnetism. Our theory allows introduction of the cluster network connectivity as a controllable parameter. Two types of inner cluster geometry are considered: triangular and tetrahedral. The theory was developed for general, nonuniform intracluster interactions, but in the present paper the results presented correspond to uniform, antiferromagnetic (AF) intraclusters interaction J_{0}/J. The clusters are represented by nodes on a finite connectivity random graph, and the intercluster interactions are randomly Gaussian distributed. The graph realizations are treated in replica theory using the formalism of order parameter functions, which allows one to calculate the distribution of local fields and, as a consequence, the relevant observable. In the case of triangular cluster geometry, there is the onset of a classical spin liquid state at a temperature T^{*}/J and then, a cluster spin glass (CSG) phase at a temperature T_{/}J. The CSG ground state is robust even for very weak disorder or large negative J_{0}/J. These results does not depend on the network connectivity. Nevertheless, variations in the connectivity strongly affect the level of frustration f_{p}=-Θ_{CW}/T_{f} for large J_{0}/J. In contrast, for the nonfrustrated tetrahedral cluster geometry, the CSG ground state is suppressed for weak disorder or large negative J_{0}/J. The CSG boundary phase presents a reentrance which is dependent on the network connectivity.

Ising spin glass in a random network with a Gaussian random field

We investigate thermodynamic phase transitions of the joint presence of spin glass (SG) and random field (RF) using a random graph model that allows us to deal with the quenched disorder. Therefore, the connectivity becomes a controllable parameter in our theory, allowing us to answer what the differences are between this description and the mean-field theory i.e., the fully connected theory. We have considered the random network random field Ising model where the spin exchange interaction as well as the RF are random variables following a Gaussian distribution. The results were found within the replica symmetric (RS) approximation, whose stability is obtained using the two-replica method. This also puts our work in the context of a broader discussion, which is the RS stability as a function of the connectivity. In particular, our results show that for small connectivity there is a region at zero temperature where the RS solution remains stable above a given value of the magnetic field no matter the strength of RF. Consequently, our results show important differences with the crossover between the RF and SG regimes predicted by the fully connected theory.

Equations-of-motion approach to the spin- 1/2 Ising model on the Bethe lattice

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 functions are written in terms of a finite set of parameters to be self-consistently determined. A procedure is developed that allows us to exactly fix the unknown parameters in the case of a Bethe lattice with any coordination number z. Nonlocal correlation functions up to four points are also provided together with a study of the relevant thermodynamic quantities.

Exact phase diagrams for an Ising model on a two-layer Bethe lattice

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 and find that when /J3/-->0, DeltaT(c) identical with[T(c)(J3)-T(c)(0)]/T(c)(0) approximately [J3]/J(1)/(1/psi), where T(c)(J3) is the phase-transition temperature for the system with interlayer coupling constant J3 and the shift exponent psi is 1 for J(1)=J(2) and is 0.5 for J1 not equal to J2. Such results are consistent with predictions of a scaling theory. We also derive equations for DeltaT(c) when /J3/ approaches infinity.