Princeton University Donald R. Hamilton Colloquium Series

Abstract: 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: linear polarizability in insulators, linear dc conductivity in met- als, Born effective charges in insulators and in metals, Hall conductivity (quantized and other- wise), and quantized Faraday charges in electrolytes. Remarkably, the adiabatic limit is well defined even in metals, despite the absence of a spectral gap therein. For all of the above ob- servables 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 (prolix) band-structure formulas for DFT implementations is straightforward.