Nonhermitian Green's function formalism for metal-insulator transition

Landau formulated a concept of Fermi liquid (FL) by adiabatically turning on interaction in a non-interacting Fermi-gas. Such evolution results in a one-to-one correspondence between excitations in the Fermi gas and the quasiparticles in the FL. Quasiparticle picture in the FL, however, breaks down when the self-energy term in the Green's function becomes non-analytic. Understanding the non-trivial physics beyond FL theory posits an open challenge in condensed matter physics. To deal with the non-analyticity of the self-energy of the Green's function, we have studied Green's function formalism for a general non-hermitian Hamiltonian in a bi-orthogonal basis. The non-hermitian term is finally set to zero to achieve a general Green's function formalism for the hermitian system. While we restore the well-studied structure of the retarded and advanced Green's function for a hermitian system by imposing an extra condition of the analyticity of the Green's function, we showed that such constraints on analyticity are no longer valid when FL theory breaks down and we get a different perturbative expansion of the Green's function. In this process, we have constructed an order parameter for the breaking down of the FL theory. For the particle-hole symmetric Hubbard model, this order parameter coincides with the order parameter of the Hubbard-I approximation.