Geometrical Observables of the Electronic Ground State

In a crystalline material the geometrical observables obtain either as Brillouin-zone integrals (in insulators), or as Fermi-volume integrals (in metals). In two cases the integrand is gauge-dependent and the bulk observables are defined modulo a "quantum": these are polarization and the "axion" term in magnetoelectric response. For bounded samples, the actual values of these two observables depend on the details of the sample termination. Remarkably, four other cases belong to a very different class, in that the integrand is gauge invariant, and the observable is independent of the sample boundary. They are: Drude weight (in metals), orbital magnetization, gauge-invariant quadratic spread (in insulators), and anomalous Hall conductivity. In our work we first provide a common k-space formalism for the four observables; then we also show that all of them admit a dual formulation in r space, and are local in character. Our local formulation allows to address the above four observables in noncrystalline samples, as well as in bounded and/or inhomogeneous samples. Simulations on paradigmatic cases validate our approach.
A. Marrazzo & R. Resta, PRL 116, 137201 (2016); PRB 95, 121114R (2017);
R. Resta, arXiv:1703.00712.