F.A. da Costa

First-order transition in a spin-glass model

We consider a generalization of the infinite-range Sherrington-Kirkpatrick spin-glass model with arbitrary spin S and the inclusion of crystal-field effects. For integer S, replica-symmetric calculations have shown the presence of both continuous and discontinuous transitions and a tricritical point. For S=1, we report a detailed numerical analysis of the replica-symmetric solutions. We locate the first-order boundary and clarify some inconsistencies of the previous analyses. Some analytic asymptotic expansions are used to support the numerical findings.

Effects of random biquadratic couplings in a spin-1 spin-glass model

Abstract:A spin-1 model, appropriated to study the competition between bilinear (JijSiSj) and biquadratic (KijSi2Sj2) random interactions, both of them with zero mean, is investigated. The interactions are infinite-ranged and the replica method is employed. Within the replica-symmetric assumption, the system presents two phases, namely, paramagnetic and spin-glass, separated by a continuous transition line. The stability analysis of the replica-symmetric solution yields, besides the usual instability associated with the spin-glass ordering, a new phase due to the random biquadratic couplings between the spins.

Mean-field solution of the Blume-Capel model under a random crystal field

Abstract In this work we investigate the Blume–Capel model with infinite-range ferromagnetic interactions and under the influence of a quenched disorder – a random crystal field. For a suitable choice of the random crystal field the model displays a wealth of multicritical behavior, continuous and first-order transition lines, as well as re-entrant behavior. The resulting phase diagrams show a variety of topologies as a function of the disorder parameter p . A comparison with recent results on the Blume–Capel model in random crystal field is discussed.

The anisotropic Ashkin–Teller model: a renormalization group study

The two-dimensional ferromagnetic anisotropic Ashkin–Teller model is investigated through a real-space renormalization-group approach. The critical frontier, separating five distinct phases, recovers all the known exact results for the square lattice. The correlation length (νT) and crossover (φ) critical exponents are also calculated. With the only exception of the four-state Potts critical point, the entire phase diagram belongs to the Ising universality class.

The random field Blume-Capel model revisited

Abstract We have revisited the mean-field treatment for the Blume-Capel model under the presence of a discrete random magnetic field as introduced by Kaufman and Kanner (1990). The magnetic field ( H ) versus temperature ( T ) phase diagrams for given values of the crystal field D were recovered in accordance to Kaufman and Kanner original work. However, our main goal in the present work was to investigate the distinct structures of the crystal field versus temperature phase diagrams as the random magnetic field is varied because similar models have presented reentrant phenomenon due to randomness. Following previous works we have classified the distinct phase diagrams according to five different topologies. The topological structure of the phase diagrams is maintained for both H - T and D - T cases. Although the phase diagrams exhibit a richness of multicritical phenomena we did not found any reentrant effect as have been seen in similar models.

Reentrant behavior in the transverse ordering of m-vector spin glasses

Abstract The infinite-range-interaction m -vector spin glass, in the presence of a uniaxial anistropy, as well as of an external magnetic field (both along the same axis), is studied through the replica method. It is shown that the Gabay-Toulouse line, which signals the ordering of the transverse degrees of freedom in the magnetic-field versus temperature plane, exhibits a reentrant behavior. Possible connections with experimental observations are discussed.

First-order transitions and triple point on a random p-spin interaction model

The effects of competing quadrupolar- and spin-glass orderings are investigated on a spin-1 Ising model with infinite-range random p -spin interactions. The model is studied through the replica approach and a phase diagram is obtained in the limit p . The phase diagram, obtained within replica-symmetry breaking, exhibits a very unusual feature in magnetic models: three first-order transition lines meeting at a common triple point, where all phases of the model coexist.

Zero-temperature TAP equations for the Ghatak-Sherrington model

The zero-temperature TAP equations for the spin-1 Ghatak-Sherrington model are investigated. The spin-glass energy density (ground state) is determined as a function of the anisotropy crystal field D for a large number of spins. This allows us to locate a first-order transition between the spin-glass and paramagnetic phases within a good accuracy. The total number of solutions is also determined as a function of D.Abstract:The zero-temperature TAP equations for the spin-1 Ghatak-Sherrington model are investigated. The spin-glass energy density (ground state) is determined as a function of the anisotropy crystal field D for a large number of spins. This allows us to locate a first-order transition between the spin-glass and paramagnetic phases within a good accuracy. The total number of solutions is also determined as a function of D.

The infinite-range mixed-spin ising glass

We use the replica method to investigate the behavior of an infinite-range mixed-spin (spin-12 and spin-1) Ising glass in the presence of a crystal field. The properties of the model are described in terms of two sets of order parameters. We obtain the replica-symmetric spin-glass solutions and investigate their stability properties. The phase diagram displays just a simple line of continuous phase transitions. We use Parisis scheme to break the replica symmetry in the neighborhood of the transition line.

Chaotic behavior of a spin-glass model on a Cayley tree

We investigate the phase diagram of a spin-1 Ising spin-glass model on a Cayley tree. According to early work of Thompson and collaborators, this problem can be formulated in terms of a set of nonlinear discrete recursion relations along the branches of the tree. Physically relevant solutions correspond to the attractors of these mapping equations. In the limit of infinite coordination of the tree, and for some choices of the model parameters, we make contact with findings for the phase diagram of more recently investigated versions of the Blume–Emery–Griffiths spin-glass model. In addition to the anticipated phases, we numerically characterize the existence of modulated and chaotic structures.