Chern number and orbital magnetization in ribbons, polymers, and single-layer materials

The modern theory of orbital magnetization addresses crystalline materials at the noninteracting level: Therein the observable is the $\mathbf{k}$-space integral of a geometrical integrand. Alternatively, magnetization admits a local representation in $\mathbf{r}$ space, i.e., a ``density,'' which may address noncrystalline and/or inhomogeneous materials as well; the Chern number admits an analogous density. Here, we provide the formulation for ribbons, polymers, nanowires, and single-layer materials, where both $\mathbf{k}$-space and $\mathbf{r}$-space integrations enter the definition of the two observables.