Machine learning approach for longitudinal spin fluctuation effects in bcc Fe atTcand under Earth-core conditions

We propose a machine learning approach to predict the shapes of the longitudinal spin fluctuation (LSF) energy landscapes for each local magnetic moment. This approach allows the inclusion of the effects of LSFs in, e.g., the simulation of a magnetic material with ab initio molecular dynamics in an effective way. This type of simulation requires knowledge of the reciprocal interaction between atoms and moments, which, in principle, would entail calculating the energy landscape of each atom at every instant in time. The machine learning approach is based on the kernel ridge regression method and developed using bcc Fe at the Curie temperature and ambient pressure as a test case. We apply the trained machine learning models in a combined atomistic spin dynamics and ab initio molecular dynamics (ASD-AIMD) simulation, where they are used to determine the sizes of the magnetic moments of every atom at each time step. In addition to running an ASD-AIMD simulation with the LSF machine learning approach for bcc Fe at the Curie temperature, we also simulate Fe at temperature and pressure comparable to the conditions at the Earth's inner solid core. The latter simulation serves as a critical test of the generality of the method and demonstrates the importance of the magnetic effects in Fe in the Earth's core despite its extreme temperature and pressure.