Analysis of the Symmetry Properties of Large Periodic Magnetic Systems, to Reduce the Computation Time of the Calculation of the Magnetostatic Dipolar Energy

The computational effort to calculate the magnetostatic dipolar energy, MDE, of a periodic cell of N magnetic moments is an O(N
²
) task. Compared with the calculation of the Exchange and Zeeman energy terms, this is the most computationally expensive part of the atomistic simulations of the magnetic properties of large periodic magnetic systems. To reduce the computational effort, the traditional Ewald method to calculate the MDE of periodic magnetic systems has been analyzed. The detailed analysis reveals that, for certain types of periodic systems, there are many matrix elements of the Ewald method identical to another elements, due to symmetry properties of the periodic systems. Computation timing experiments of the MDE of large systems, such as Ni fcc nanowires up to 31500 magnetic moments in the periodic cell, have been carried out and they show that the number of matrix elements that should be calculated is approximately equal to N, instead of N
²
/2 if these symmetries are used, and that the computation time decreases in an important amount. The time complexity of the analysis of the symmetries is O(N
³
), which increases the time complexity of the traditional Ewald method and is in contrast with the computation timing experiments. This is explained by the fact that the MDE is a very small energy and therefore, the usual required precision of the calculation of the MDE is so high, about 10
^-6
eV/cell, that the calculations of large periodic magnetic systems are very expensive and the use of the symmetries reduces, in practical terms, the computation time of the MDE in a significant amount, in spite of the increase of the time complexity.