Longitudinal integration measure in classical spin space and its
application to first-principle based simulations of the high temperature
magnetism of Fe and Ni
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.Comment: Journal of Magnetism and Magnetic Materials Available online 12 April
2018 In Press, Accepted Manuscrip
