Sum rules for density and particle excitations in a superfluid of charged bosons

We derive and discuss sets of sum rules relating to the density fluctuation operator and to the single-particle creation and annihilation operators in a superfluid of charged bosons. The physical interpretation of the particle-particle and particle-density sum rules hinges on the single-particle excitation spectrum at long wavelengths having a gap equal to the plasma frequency in the presence of the Bose-Einstein condensate. This spectral property is shown to follow from the Hugenholtz-Pines relation for the chemical potential in terms of the half-diagonal two-body density matrix. Data on this density matrix are obtained by quantum Monte Carlo methods and are used to check the self-consistency between the Hugenholtz-Pines relation and the value of the chemical potential calculated from the ground-state energy. We also tabulate the contributions from the plasmon and from multiparticle excitations to various matrix elements and sum rules at long wavelengths.