Abstract — Several physical observables of materials have their theoretical root in geometrical properties of the electronic ground state. To start with, I will outline the modern theory of the insulating state, which addresses all kinds of insulators (band, Mott, Anderson...), discriminating them from metals by means of a very simple geometrical property of their ground state. Next I will specialize to either band insulators or band metals, focusing on some observables which stem from the geometry of the occupied manifold in reciprocal space. Nowadays the most popular geometrical observable is electrical polarization (for insulators), whose expression is a Berry phase: the k-space integrand is gauge-dependent and the bulk observable is defined only modulo a "quantum". Some other observables, instead, obtain from a gauge-invariant k-space integrand and are free from any "quantum" ambiguity: these include orbital magnetization and anomalous Hall conductivity (both defined for either metals or insulators). Other observables in this class will be discussed as well. Recent work has shown that orbital magnetization and anomalous Hall conductivity also admit a dual representation in coordinate space, and can be evaluated for a bounded sample—even noncrystalline—with square-integrable orbitals (where k-space doesn’t make any sense)
 A. Marrazzo and R. Resta, Irrelevance of the boundary on the magnetization of metals, Phys. Rev. Lett. 116, 137201 (2016).
 A. Marrazzo and R. Resta, Locality of the anomalous Hall conductivity, Phys. Rev. B 95, 121114(R) (2017).
 R. Resta, Geometrical meaning of the Drude weight and its relationship to orbital magnetization, arXiv:1703.00712 (2017).
About the speaker — Raffaele Resta is a retired professor of physics, presently senior research associate with CNR (Italy). Previously he served in Trieste, first at SISSA (1983-1994) and then at the University of Trieste (1995-2017); he has also been long-term visitor at EPFL several times. Since the beginning of professional life his main interest has been in the theory of materials, using a variety of approaches, from analytical theories and models to firstâprinciple computations. Since the birthdate (about 1980) of the modern computational theory of materials, his mainstream research activity has been in this area, working both at the development of new methods and at actual computations. He has made crucial advances in the understanding of macroscopic polarization, orbital magnetization, magnetoelectric couplings, flexoelectricity, and the nature of the insulating state. He is the author of several review papers on the above topics. He is a fellow of the American Physical Society, and a former (2002-08) Divisional Associate Editor for Physical Review Letters.
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