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

• Peculiarities of 1D and 2D Ising model with competing interactions are studied.
• All possible structures, frustration points and frustration fields are found.
• At appropriate choice of a model, magnetic properties in 1D and 2D cases are alike.
• Similarities and dissimilarities between considered models are discussed.

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.