Fermi-liquid relations for arbitrary clusters of Anderson impurities

We consider an arbitrary cluster of interacting single-orbital Anderson impurities. A Fermiliquid theory is derived by making use of the Ward identities corresponding to the conservation of the total charge and the total spin. The specific heat and the charge and spin susceptibilities are related by Fermi-liquid parameters, and the Korringa relations for the charge-lattice and spin-lattice relaxation are derived. The results are valid for arbitrary (finite or infinite) clusters of impurities in the absence of long-range order, e.g., for isolated impurities and for the Anderson lattice. The results simplify in the Kondo limit where we obtain the relation $2\gamma =\left(\frac{{\pi }^2}{3}\right){\chi }_s$ for any spatial configuration of spin-½ Kondo impurities.