Exact many-body sum rule for the magneto-optical spectrum of solids

Using a many-body formulation we show that the integral over the imaginary part of the off-diagonal optical conductivity gives---at zero temperature---exactly the many-electron center of mass angular moment in the volume, i.e., ${{∫}}₀^{{∞}}Im[{{σ}}xy({ω})]d{ω}{∝}〈(R×{}P{)}z〉,$ with $R$ the center of mass coordinate and $P$ the total relativistic momentum. At elevated temperatures the sum rule is no longer exactly fulfilled, but the temperature-related correction is estimated to be very small. In a single-particle description the sum rule leads to the total orbital moment in the volume plus an additional term.