Critical metal-insulator transition and divergence in a two-particle irreducible vertex in disordered and interacting electron systems

We use the dynamical mean-field approximation to study singularities in the self-energy and a two-particle irreducible vertex induced by the metal-insulator transition of the disordered Falicov-Kimball model. We set general conditions for the existence of a critical metal-insulator transition caused by a divergence of the imaginary part of the self-energy. We calculate explicitly the critical behavior of the self-energy for the symmetric and asymmetric disorder distributions. We demonstrate that the metal-insulator transition is preceded by a pole in a two-particle irreducible vertex. We show that unlike the singularity in the self-energy the divergence in the irreducible vertex does not lead to non-analyticities in measurable physical quantities. We reveal universal features of the critical metal-insulator transition that are transferable also to the Mott-Hubbard transition in the models of the local Fermi liquid.