Magnetic interactions in strongly correlated systems: spin and orbital contributions

摘要

We present a technique to map an electronic model with local interactions (ageneralized multi-orbital Hubbard model) onto an effective model of interactingclassical spins, by requiring that the thermodynamic potentials associated tospin rotations in the two systems are equivalent up to second order in therotation angles. This allows to determine the parameters of relativistic andnon-relativistic magnetic interactions in the effective spin model in terms ofequilibrium Green's functions of the electronic model. The Hamiltonian of theelectronic system includes, in addition to the non-relativistic part,relativistic single-particle terms such as the Zeeman coupling to an externalmagnetic fields, spin-orbit coupling, and arbitrary magnetic anisotropies; theorbital degrees of freedom of the electrons are explicitly taken into account.We determine the complete relativistic exchange tensors, accounting foranisotropic exchange, Dzyaloshinskii-Moriya interactions, as well as additionalnon-diagonal symmetric terms (which may include dipole-dipole interaction). Ourprocedure provides the complete exchange tensors in a unified framework,including previously disregarded features such as the vertices of two-particleGreen's functions and non-local self-energies. We do not assume any smallnessin spin-orbit coupling, so our treatment is in this sense exact. Finally, weshow how to distinguish and address separately the spin, orbital andspin-orbital contributions to magnetism.