We introduce a one dimensional Ising model with two competing interactions: nearest neighbor random couplings +/-J with equal probability and a positive infinite range coupling Lambda. At low temperature T the model exhibits a first order phase transition between a ferromagnetic state (with magnetization m(1) = 1 at T = 0) and a << ferrimagnetic >> state (with m(2) = 2/3 at T = 0), when the disorder strength J/Lambda is increased. For 5/12 < J/Lambda < 1, a whole spectrum of ferrimagnetic ground states with magnetization m(n) = 2/(n + 1) (n = 2, ..., infinity) is present while for J/lambda > 1 the ground state is given by a trivial one dimensional spin glass with m = 0. The main qualitative features of the model can be described by a simplified annealed model where the random couplings can arrange themselves to minimize free energy with the constraint that the number of positive couplings is fixed by the law of large numbers in the thermodynamic limit. This model is exactly solved at all temperatures and the diagram of phase is calculated.
Abstract Phase transitions and critical and thermodynamic properties of the three-dimensional antiferromagnetic Ising model on a layered triangular lattice with variable interlayer exchange interaction are studied by the replica algorithm of the Monte Carlo method. The studies are carried out for the ratios of the intralayer J1 and interlayer J2 exchange interactions in the range of r = J2/J1 = 0.01–1.0. It is established that a second-order phase transition is observed in the considered r interval. Using the finite size scaling theory, the static critical exponents of the heat capacity α, susceptibility γ, order parameter β, correlation radius ν, and Fisher index η are calculated. It is shown that the universality class of the critical behavior of this model is preserved in the interval of 0.05 < r ≤ 1.0. It was found that with a further decrease in the r value, a crossover from the three-dimensional critical behavior to the quasi-two-dimensional one is observed.
The Monte Carlo method was used to study the phase transitions and thermodynamic properties of the three-dimensional antiferromagnetic Ising model on a layered triangular lattice at various values of interlayer exchange interaction. A number of investigations have been carried out for the ratios of the intralayer (J1) and interlayer (J2) exchange interactions within the range of r = J2/J1 = 0.01–1.0. The nature of the phase transitions was studied using the fourth-order Binder cumulant method and histogram data analysis. It has been shown that in the considered range r, the transition from a disordered phase to an ordered phase is a second-order phase transition.
In this study, we explored the magnetic coupling in a multilayer system consisting of thin layers separated by a distance
d. We have employed Monte Carlo (MC) simulations to calculate the thermodynamic quantities such as the magnetization per spin
mLμ, magnetic susceptibility
χLμ, and the reduced fourth-order Binder cumulant
ULμ as a function of temperature
T and for several values of lattice size
L. These quantities were obtained for each layer (
μ=l), for the bulk system (
μ=b), as a function of interlayer distance
d and the parameter
α that defines the scale length of the exponential decay of the interaction. Furthermore, we applied the finite-size scaling theory to calculate the critical exponents. Our results reveal that the system exhibits 2D Ising exponents when the layers are sufficiently separated. On the other hand, in the compact limit, where
d equals the distance between two adjacent sites within the same layer, our bulk results show that the system exhibits 3D Ising critical exponents, provided that the number of layers
Lz increases proportionally to the layer sizes. Even with the separation increasing up to
d=2.0, the layers are so correlated that the set of critical exponents retains the values of 3D critical exponents. However, when the number of layers
Lz remains fixed, even in the compact limit with periodic boundary conditions, only the exponent
β aligns closely with the predicted literature values, on the other hand, the other exponents show significant deviations.
Abstract In this work, the Ising model on a trilayer Bethe lattice is studied in terms of recursion relations for which the nearest‐neighbor spins of each layer are coupled ferromagnetically and the adjacent spins of the nearest‐neighbor layers are coupled either ferromagnetically or antiferromagnetically depending on the sign of the bilinear interactions for given values of the coordination number q. Besides the ground‐state phase diagram, the phase diagrams of the model are also calculated for given values of the system parameters. It is found that the model presents both first‐ and second‐order phase transitions for all values of q. In order to construct the phase diagrams, we have also studied the thermal and interlayer coupling constant changes of the layer magnetizations, the spin–spin correlation functions between the nearest‐neighbor spins of the adjacent layers and the free energy of the model. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
Ising models, with quenched bond disorder along one direction on a square lattice, are studied by the transfer-matrix method. The exact criticality condition is rederived, and extended to an arbitrary variation, in magnitude and/or sign, of the interactions from layer to layer. With competing interactions, a phase diagram is obtained, with a zero-temperature spin-glass point. We consider
2n different nearest-neighbor couplings layered and repeated periodically along one direction. The phase boundaries, including zero-temperature bicritical points, the order parameters, and the ground-state degeneracies are illustrated by a detailed discussion for
n=2. Closed-form free energies are given for
n=2and3. The unique criticality condition is derived for any
n, the limit $n\ensuremath{\rightarrow}\ensuremath{\infty}$ corresponding to quenched disorder. Our extended result is made possible by a similarity transformation in the free-fermion representation. A conjecture emerges for the criticality condition of randomly layered ferromagnetic
q-state Potts models.