Phase diagram of the two-dimensional Ising model with random competing interactions

Abstract

An Ising model with ferromagnetic nearest-neighbor interactions J 1 (J 1 > 0) and random next-nearest-neighbor interactions [+J 2 with probability p and −J 2 with probability (1 − p); J 2 > 0] is studied within the framework of an effective-field theory based on the differentialoperator technique. The order parameters are calculated, considering finite clusters with n = 1, 2, and 4 spins, using the standard approximation of neglecting correlations. A phase diagram is obtained in the plane temperature versus p, for the particular case J 1 = J 2 , showing both superantiferromagnetic (low p) and ferromagnetic (higher values of p) orderings at low temperatures.Many magnetic compounds are well-described in terms of theoretical models characterized by a competition of nearest-neighbor and next-nearest-neighbor interactions. As typical examples, one has Eu x Sr 1−x S [1, 2] and Fe x Zn 1−x F 2 [3], which may present various low-temperature magnetic orderings, depending on its parameters, like the strength of these interactions and the concentration of magnetic ions x. The simplest model for such systems consists of an Ising model with competing uniform interactions, ferromagnetic J 1 (or antiferromagnetic −J 1 ) (J 1 > 0) on nearest-neighbor, and antiferromagnetic −J 2 (J 2 > 0) on next-nearest-neighbor spins (to be denoted hereafter as J 1 − J 2 Ising model).The J 1 −J 2 Ising model on a square lattice has attracted the attention of many workers, being investigated through several approximation methods ]. The various possible phases are the paramagnetic (P), for high temperatures, whereas for low temperatures one may have the ferromagnetic (F), antiferromagnetic (AF) and superantiferromagnetic (SAF) phases; the later is characterized by alternate ferromagnetic rows (or columns) of oppositely oriented spins. In the absence of a magnetic field, it can be shown that one has a symmetry with respect to the sign of the nearestneighbor interactions, i.e., the ferromagnetic and antiferromagnetic states are equivalent [5].At zero temperature (ground state), one has two ordered states depending on the value of the frustration parameter α = (J 2 /J 1 ), namely, the F (0 < α < 1/2) (in an equivalent way, an AF state occurs instead of the F one, if the sign of nearest-neighbor interactions is changed) and the SAF (α > 1/2) phases. On the other hand, its phase diagram for finite temperatures has been the object of some controversies. A continuous critical frontier between the P and F phases, for 0 < α < 1/2, with the critical temperature T c (α) decreasing and approaching zero for α → 1/2, is well-accepted nowadays. However, some characteristics of this phase diagram for α > 1/2 are polemic; in particular, the presence of a first-order transition between the P and SAF phases, as well as a tricritical point characterized by the coordinates (α t , T t ), beyond which a continuous critical frontier occurs, has been the object of some debate. Most of the works indicate that within the range 1/2 < α < α t one should get a first-order pha...

Tuning a random-field mechanism in a frustrated magnet

“…In Ref. 35, this likely stems from the fact that the generalized mean-field theory does not consider the possibility of random-field physics at all. In the Wang-Landau study in Ref.…”

Effective field theory in larger clusters – Ising model

“…We mention that most of the studies in related literature concerns with 1 or 2-spin cluster, although limited works using 4-spin cluster have also been presented, such as Ref. [19].…”

“…For instance, 2-spin cluster (EFT-2) [18] and 4-spin cluster (EFT-4) [19] have been successfully applied to the Ising systems. But, to the best of our knowledge, there is no general formulation for EFT-N given.…”

“…On the other hand, larger clusters for obtaining critical properties of the Ising model for several lattices have been used. For instance, 2-spin cluster (EFT-2) [18] and 4-spin cluster (EFT-4) [19] have been successfully applied to the Ising systems. But, to the best of our knowledge, there is no general formulation for EFT-N given.…”

Bicritical Central Point of Ising Model Phase Diagram

“…The study of frustrated magnetic systems always arouses the interest [1][2][3][4][5][6][7]. It is known that the nature of the ordering at low temperatures regime of these materials depends on the concentration of the magnetic species and on the relative magnitude of competing energy scales [8,9].…”