Orbital Magnetization in Solids.

Raffaele Resta

Dipartimento di Fisica, Università di Trieste, Italy, and DEMOCRITOS National Simulation Center,

IOM-CNR, Trieste, Italy

Spontaneous polarization P and magnetization M are basic topics in electrostatics and magnetostatics. Despite this a microscopic understanding of what these properties are is relatively recent: 1992-3 for P, and 2005-6 for the orbital contribution to M [1]. Textbooks typically attempt microscopic definitions in terms of the dipole moment per cell: this is only correct for the spin contribution to M, but is incorrect for both P and for the orbital contribution to M. The microscopic quantities well defined are rho(micro)(r) and j(micro)(r), but their bulk value does not determine P and M (we use the symbol M for the orbital term only here and in the following). The dipole moment—either electric or magnetic—of a macroscopic sample is clearly dominated by surface contributions; therefore according to the naive definitions (total dipole divided by the sample volume) P and M are apparently not bulk properties.

The modern theories express both P and M of a crystalline system as k integrals of Bloch-orbital matrix elements [1]: these are clearly “bulk” by definition. The ground state of a solid is uniquely determined by the one-particle density matrix P(r, r′). It is therefore desirable to express P and M in terms of P(r, r′) directly in r space: I will show that this is not possible in the case of P, while for M we found an explicit local magnetization formula in terms of P(r, r′); simulations on a model Hamiltonian validate our theory [2]. At variance with the k space formula, our r space formula forM applies with no major changes to noncrystalline systems (disordered and/or macroscopically inhomogenous).

[1] R. Resta, J. Phys.: Condens. Matter 22 123201 (2010).

[2] R. Bianco and R. Resta, Phys. Rev. Lett. 110, 087202 (2013).