Longitudinal integration measure in classical spin space and its application to first-principle based simulations of the high temperature magnetism of Fe and Ni

TL;DR

This paper derives a new integration measure for classical spin space that improves the modeling of temperature-dependent magnetism in Fe and Ni, incorporating longitudinal spin fluctuations from first-principles calculations.

Contribution

It introduces a fundamental derivation of the integration measure in classical spin space considering quantum limits, enhancing the accuracy of finite-temperature magnetic simulations.

Findings

The derived measure accounts for the proportionality of quantum states to spin amplitude.

Application to Fe and Ni yields more accurate Curie temperature predictions.

Comparison shows the new measure outperforms previous scalar and vector measures.

Abstract

The classical Heisenberg type spin Hamiltonian is widely used for simulations of finite temperature properties of magnetic metals often using parameters derived from first principles calculations. In itinerant electron systems, however, the atomic magnetic moments vary their amplitude with temperature and the spin Hamiltonian should thus be extended to incorporate the effects of longitudinal spin fluctuations (LSF). Although the simple phenomenological spin Hamiltonians describing LSF can be efficiently parameterized in the framework of the constrained Local Spin Density Approximation (LSDA) and its extensions, the fundamental problem concerning the integration in classical spin space remains. It is generally unknown how to integrate over the spin amplitude. Two intuitive choices of integration measure have been used up to date; the Murata-Doniach scalar measure and the simple three dimensional vector measure. Here we derive the integration measure by considering a classical limit of the quantum Heisenberg spin Hamiltonian under conditions leading to the proper classical limit of the commutation relations for all values of the classical spin amplitude and calculate the corresponding ratio of the number of quantum states. We show, that the number of quantum states corresponding to the considered classical spin amplitude is proportional to this amplitude and thus a non-trivial integration measure must be used. We apply our results to the first-principles simulation of the Curie temperatures of the two canonical ferromagnets bcc Fe and fcc Ni using a single-site LSF Hamiltonian with parameters calculated in the LSDA framework in the Disordered Local Moment approximation and a fixed spin moment constraint. In the same framework we compare our results with those obtained from the scalar and vector measures.