Towards a Theory of Surface Orbital Magnetization

The bulk values of macroscopic quantities like electric polarization and orbital magnetization for crystalline systems of independent electrons may be explicitly computed using k-space integrals of quantities involving Berry connections and curvatures of Bloch functions. In recent years, attention has focused on expressing the same quantities locally via one-particle density matrices. Such decompositions may be exploited to define surface dependent analogues of bulk quantities; an example is the anomalous Hall conductivity (AHC) on the surface of a bulk. In this talk, we explore the possibility of defining a surface orbital magnetization on the boundary of a three-dimensional bulk system. We address this question via a combination of formal arguments and test calculations on a layered realization of the Haldane model. While we focus here on topologically trivial materials, we also speculate on the connections to topological ones, e.g., the relation between orbital magnetization and AHC on the surface of a material with a non-zero Chern-Simons axion coupling.