On the density-density critical indices in interacting Fermi systems

The behaviour of correlation functions of d = 1 interacting fermionic systems is determined by a small number of critical indices. We prove that one of them is exactly zero. As a consequence, the behavior of the Fourier transform of the density-density correlation at zero momentum is qualitatively unaffected by the interaction, contrary to what happens at ±2p̃F, if p̃F is the Fermi momentum. The result is obtained by implementing Ward identities in a Renormalization Group approach.