Compressibility and density response function in disordered systems

We give a detailed study on the compressibility and the density response function in a disordered system which has a single-band or single-valley structure with a spherical energy surface, in the absence of external magnetic fields. In analysis we use a diagrammatic method which gets the benefit of Finkelsteins renormalization scheme and can be applied easily to cases where there are interactions between particles and other elementary excitations. It is shown that in obtaining the relation between the compressibility and the density response function in the static and the long-wavelength limits, 0 and q0, we need diagrams which are different from familiar electron-hole bubble diagrams and which have not been discussed so far. The following fact is ascertained: The interaction among particles involving thermal diffusion processes does not contribute to the compressibility as a whole, but gives a significant influence on the density response function in the dynamical case, where both the external frequency and wave vector are finite, provided that a typical excitation energy due to the thermal fluctuation is much larger than that due to the external disturbance. Furthermore, it is shown that as a result of the particle number conservation law there should exist no term which prevents the divergence of the electron-hole diffusion propagator in the limits 0 and q0.