Orbital Magnetization in Periodic Solids

A complete description of magnetism in solids requires not only the spin degrees of freedom, but also the "orbital magnetization." Despite the recent surge of interest in magnetic materials, it is quite surprising that the theory of orbital magnetization has remained in a condition similar to that of the polarization before the early 1990s, when the problem of computing finite polarization changes was solved [R.D. King-Smith and D. Vanderbilt, Phys. Rev. B 47, 1651 (1993)]. The essential difficulty, that the matrix elements of the position operator "r" are not well-defined in the Bloch representation, could be overcome by reformulating the problem in the Wannier representation. In order to derive an analogous theory for the orbital magnetization, we again work in the Wannier representation and assume a periodic insulator with broken time-reversal symmetry and vanishing magnetic field. We show that a naive replacement of the dipole operator "r" by the circulation operator "r x v" in the expectation value of a bulk Wannier function gives only one contribution to the magnetization, i.e., the magnetization associated with the internal circulation of bulk-like Wannier functions. The missing contribution arises from net currents carried by the Wannier functions at the boundary of the sample. We prove that both contributions can be expressed as bulk properties in terms of Bloch functions in a gauge-invariant way [T. Thonhauser, D. Ceresoli, D. Vanderbilt, and R. Resta, Phys. Rev. Lett. 95, 137205 (2005)].