M. K. Ramazanov

Ising model on a square lattice with second-neighbor and third-neighbor interactions

We studied the phase transitions and magnetic properties of the Ising model on a square lattice by the replica Monte Carlo method and by the method of transfer-matrix, the maximum eigenvalue of which was found by Lanczos method. The competing exchange interactions between nearest neighbors J1, second J2, third neighbors J3 and an external magnetic field were taken into account. We found the frustration points and expressions for the frustration fields, at crossing of which cardinal changes of magnetic structures (translational invariance changes discontinuously) take place. A comparative analysis with 1D Ising model was performed and it was shown that the behavior of magnetic properties of the 1D model and the 2D model with J1 and J3 interactions reveals detailed similarity only distinguishing in scales of magnetic field and temperature.

Phase transitions in the antiferromagnetic ising model on a square lattice with next-nearest-neighbor interactions

The phase transitions in the two-dimensional Ising model on a square lattice are studied using a replica algorithm, the Monte Carlo method, and histogram analysis with allowance for the next-nearest-neighbor interactions in the range 0.1 ≤ r < 1.0. A phase diagram is constructed for the dependence of the critical temperature on the next-nearest-neighbor interaction. A second-order phase transition is detected in this range and the model under study.

Study of critical properties of the frustrated antiferromagnetic Heisenberg model on a triangular lattice

The replica Monte Carlo method has been applied to study critical properties of the three-dimensional frustrated antiferromagnetic Heisenberg model on a layered triangular lattice. Magnetic and chiral critical exponents for this model have been calculated within the finite-size scaling theory. The data for this model have been analyzed for the first time taking into account corrections to scaling. It has been shown that the critical behavior of the frustrated antiferromagnetic Heisenberg model on the triangular lattice differs from the critical behavior of the pure Heisenberg model and is independent of the type of interlayer exchange interaction.

Monte Carlo Investigation of Phase Transitions in the Frustrated Heisenberg Model on a Triangular Lattice

The critical properties and phase transitions of the three-dimensional frustrated antiferromagnetic Heisenberg model on a triangular lattice have been investigated using the Monte Carlo method with a replica algorithm. The critical temperature has been determined and the character of the phase transitions has been analyzed using the method of fourth-order Binder cumulants. A second-order phase transition has been found in the three-dimensional frustrated Heisenberg model on a triangular lattice. The static magnetic and chiral critical exponents of the heat capacity α, the susceptibility γ and γk, the magnetization β and βk, the correlation length ν and νk, as well as the Fisher exponents η and ηk, have been calculated in terms of the finite-size scaling theory. It has been demonstrated that the three-dimensional frustrated antiferromagnetic Heisenberg model on a triangular lattice forms a new universality class of the critical behavior.

Static critical behavior of 3D frustrated Heisenberg model on stacked triangular lattice with variable interlayer exchange coupling

Several Monte Carlo algorithms are used to examine the critical behavior of the 3D frustrated Heisenberg model on stacked triangular lattice with variable interlayer exchange coupling for values of the interlayer-to-intralayer exchange ratio R = |′/J| in the interval between 0.01 and 1.0. A finite-size scaling technique is used to calculate the static magnetic and chiral critical exponents α (specific heat), γ and γk (susceptibility), β and βk(magnetization), ν and νk(correlation length), and the Fisher exponent η. It is shown that 3D frustrated Heisenberg models on stacked triangular lattice with R > 0.05 constitute a new universality class of critical behavior. At lower R, a crossover from 3D to 2D critical behavior is observed.

Phase transitions in the antiferromagnetic Ising model on a body-centered cubic lattice with interactions between next-to-nearest neighbors

Phase transitions in the antiferromagnetic Ising model on a body-centered cubic lattice are studied on the basis of the replica algorithm by the Monte Carlo method and histogram analysis taking into account the interaction of next-to-nearest neighbors. The phase diagram of the dependence of the critical temperature on the intensity of interaction of the next-to-nearest neighbors is constructed. It is found that a second-order phase transition is realized in this model in the investigated interval of the intensities of interaction of next-to-nearest neighbors.

Phase transitions and critical properties of the frustrated Heisenberg model on a layer triangular lattice with next-to-nearest-neighbor interactions

The critical behavior of the three-dimensional antiferromagnetic Heisenberg model with nearest-neighbor (J) and next-to-nearest-neighbor (J1) interactions is studied by the replica Monte Carlo method. The first-order phase transition and pseudouniversal critical behavior of this model are established for a small lattice in the interval R = |J1/J| = 0–0.115. A complete set of the main static magnetic and chiral critical indices is calculated in this interval using the finite-dimensional scaling theory.

A study of the critical properties of the Ising model on body-centered cubic lattice taking into account the interaction of next behind nearest neighbors

The replica Monte Carlo method has been used to investigate the critical behavior of a threedimensional antiferromagnetic Ising model on a body-centered cubic lattice, taking into account interactions of the adjacent behind neighbors. Investigations are carried out for the ratios of the values of exchange interactions behind the nearest and next nearest neighbors k = J2/J1 in the range of k ∈ [0.0, 1.0] with the step Δk = 0.1. In the framework of the theory of finite-dimensional scaling the static critical indices of heat capacity α, susceptibility γ, of the order parameter β, correlation radius ν, and also the Fisher index η are calculated. It is shown that the universality class of the critical behavior of this model is kept in the interval of k ∈ [0.0, 0.6]. It is established that a nonuniversal critical behavior is observed in the range k ∈ [0.8, 1.0].

Frustrations and Phase Transitions in Low-Dimensional Magnetic Systems

We studied magnetic orderings and frustrations on 1D chain and 2D lattices: square, triangular, kagome, and hexagonal in the Ising, 3-state Potts and standard 4-state Potts models. The spins interrelate with one another via the nearest-neighbor, the next-nearest-neighbor or the third-neighbor exchange interactions and by an external magnetic field. For problem solving we mainly calculated the entropy and specific heat using the rigorous analytical solutions for maximum eigenvalue of Kramers-Wannier transfer-matrix and exploiting computer simulation, par excellence, by Wang-Landau algorithm. Whether a system is ordered or frustrated is related to the signs and values of exchange interactions. An external magnetic field may both favor the ordering of a system and create frustrations. With the help of calculations of the entropy, the specific heat and magnetic parameters, we obtained the points and ranges of frustrations, the frustration fields and the phase transition points. The results obtained also show that the same exchange interactions my either be competing or noncompeting which depends on the topology of a lattice.

Phase transitions and critical properties in the antiferromagnetic Ising model on a layered triangular lattice with allowance for intralayer next-nearest-neighbor interactions

The phase transitions (PTs) and critical properties of the antiferromagnetic Ising model on a layered (stacked) triangular lattice have been studied by the Monte Carlo method using a replica algorithm with allowance for the next-nearest-neighbor interactions. The character of PTs is analyzed using the histogram technique and the method of Binder cumulants. It is established that the transition from the disordered to paramagnetic phase in the adopted model is a second-order PT. Static critical exponents of the heat capacity (α), susceptibility (γ), order parameter (β), and correlation radius (ν) and the Fischer exponent η are calculated using the finite-size scaling theory. It is shown that (i) the antiferromagnetic Ising model on a layered triangular lattice belongs to the XY universality class of critical behavior and (ii) allowance for the intralayer interactions of next-nearest neighbors in the adopted model leads to a change in the universality class of critical behavior.