Adiabatic observables and Berry curvatures in insulators and metals

A sharp definition of what "adiabatic" means is given; it is then shown that the time-dependent expectation value of a quantum-mechanical observable in the adiabatic limit can be expressed -- in many cases -- by means of the appropriate Berry curvature. Condensed-matter observables belonging to this class include: Born effective charges in insulators and in metals, quantized Faraday charges in electrolytes, and linear dc conductivities (longitudinal and transverse). Remarkably, the adiabatic limit is well defined even in metals, despite the absence of a spectral gap therein. For all of the above observables the explicit Berry-curvature expressions are derived in a general many-body setting, which also allows for compact and very transparent notations and formulas. Their conversion into band-structure formulas in the independent-electron crystalline case is straightforward.