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., integral(0)(infinity)Im[sigma(xy)(omega)]d omega proportional to[(RXP)(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.