A. I. Proshkin

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.

Frustrations and phase transitions in the three-vertex Potts model with next-nearest-neighbor interactions on a triangular lattice

Phase transitions in two-dimensional arrays described by the three-vertex Potts model involving the interactions between magnetic moments located at the nearest-neighbor and next-nearest-neighbor sites of a triangular lattice are studied using the Monte Carlo method. The ratio of the next-nearest-neighbor and nearest-neighbor exchange constants r = J2/J1 is chosen within the 0–1 range. The analysis of the low temperature behavior of the entropy and density of states in the system, as well as of the fourth-order Binder cumulants, shows that in the range 0 ≤ r < 0.2, this model exhibits a first-order phase transition, whereas at r ≥ 0.2, frustrations arise in such a system.

Phase transitions in a two-dimensional antiferromagnetic Potts model on a triangular lattice with next-nearest neighbor interactions

Phase transitions (PTs) and frustrations in two-dimensional structures described by a three-vertex antiferromagnetic Potts model on a triangular lattice are investigated by the Monte Carlo method with regard to nearest and next-nearest neighbors with interaction constants J1 and J2, respectively. PTs in these models are analyzed for the ratio r = J2/J1 of next-nearest to nearest exchange interaction constants in the interval |r| = 0–1.0. On the basis of the analysis of the low-temperature entropy, the density of states function of the system, and the fourth-order Binder cumulants, it is shown that a Potts model with interaction constants J1 < 0 and J2 < 0 exhibits a first-order PT in the range of 0 ⩽ r < 0.2, whereas, in the interval 0.2 ⩽ r ⩽ 1.0, frustrations arise in the system. At the same time, for J1 > 0 and J2 < 0, frustrations arise in the range 0.5 < |r| < 1.0, while, in the interval 0 ⩽ |r| ⩽ 1/3, the model exhibits a second-order PT.

Exact Solution of 1D Ising Model on Linear Chain with Arbitrary Spin

We investigated the Ising model on a linear chain with arbitrary spin including interactions between nearest and next-nearest neighbors in an applied magnetic field. A series of exact solutions and formulas for frustration fields, magnetizations and entropies at these fields at T→0 are found.

Magnetocaloric effect in one-dimensional magnets

The magnetocaloric effect is studied in the context of the Kramers-Wannier transfer matrix, using exact analytical solutions for one-dimensional models of the Ising, Heisenberg, and XY models, along with planar, standard, and modified Potts models. The analytical formulas are obtained as functions of exchange parameters, values, and directions of the external magnetic field, and numerical calculations are performed.

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.

Magnetocaloric Effect and Frustrations in One-Dimensional Magnets

In this paper, we investigated the magnetocaloric effect (MCE) in one-dimensional magnets with different types of ordering in the Ising model, Heisenberg, XY-model, the standard, planar, and modified Potts models. Exact analytical solutions to MCE as functions of exchange parameters, temperature, values and directions of an external magnetic field are obtained. The temperature and magnetic field dependences of MCE in the presence of frustrations in the system in a magnetic field are numerically computed in detail.

Correlation function of one-dimensional s = 1 Ising model

The temperature behavior of the Fourier transform of the spin-correlation function has been studied in terms of the one-dimensional Ising model taking into account the interaction between the nearest neighbors in the cases of different signs of exchange interactions, magnetic field, and spin. It has been shown that, in the antiferromagnetic model, in the frustration field, the correlation function has a broad maximum and does not take on the form of a delta function as the temperature approaches zero, which indicates the absence of ordering in the system.

Diffuse Scattering on Ising Chain with Competing Interactions

We considered the Ising 1D chain in an external magnetic field taking into account the nearest and next-nearest interactions. By the method of Kramers–Wannier transfer-matrix, the rigorous analytical expression for Fourier-transform of pair spin-spin correlation function was obtained, and the temperature evolution of the scattering was analyzed for various relations of exchange parameters.

Twelve-state Potts model in a magnetic field

In this work, we have obtained an exact solution to the one-dimensional modified 12-state Potts model using the Kramers-Wannier transfer matrix with allowance for the exchange interaction between nearest neighbors in an external magnetic field. Analytical expressions have been derived for the heat capacity, magnetization, magnetic susceptibility, magnetic entropy, and magnetocaloric effect as functions of temperature, magnitude and sign of exchange interaction, and the magnitude and direction of the magnetic field. The behavior of all of these parameters has been investigated in detail using numerical methods. The possibility of applying the results obtained to explain the observed magnetic properties of real cubic magnets with a NaCl structure and easy axes oriented along the [110] crystallographic directions has been discussed.