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

TL;DR

This study investigates phase transitions and magnetic properties of a 2D Ising model on a square lattice with multiple neighbor interactions using Monte Carlo and transfer-matrix methods, revealing frustration points and structural changes.

Contribution

It introduces a detailed analysis of frustration points and magnetic structure transitions in the 2D Ising model with second and third neighbor interactions, comparing with 1D behavior.

Findings

Identification of frustration points and fields

Discontinuous changes in magnetic structures

Similarity between 1D and 2D models in certain regimes

Abstract

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 $J_{1}$, second $J_{2}$, third neighbors $J_{3}$ 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 $J_{1}$ and $J_{3}$ interactions reveals detailed similarity only distinguishing in scales of magnetic field and temperature.