Perturbation Theory and Singularities in the 2D Hubbard Model

Abstract

The two-dimensional Hubbard model is studied in the region of low electron density. In the second order perturbation with respect to Coulomb interaction, U, the imaginary part of the electron self-energy at the Fermi momentum, Im∑(k _F , ɛ + iη), has been known to have the singularity of the form ɛ ² ln |ɛ| where both regions of q ~ 0 and 2k _F , q being the total momentum of two interacting electrons, contribute to this singularity. In the higher order of U treated in the t-matrix approximation, only contributions from q ~ 2k _F survive. Next the analytical properties of the t-matrix have been studied and we found that all the poles of the t-matrix are located on the real axis of the complex energy variable and that the t-matrix is analytic in the upper half plane. By this finding it is concluded that Re∑(k _F , ɛ + iη), ~ ɛ, and then electrons in 2d-Hubbard model are a Fermi liquid at least in the region of low density. The phase shift δ(2k _F +q, ω), with k _F being the Fermi wave vector, in the limit of q = 0 and ω = 0 is found to depend strongly on the ratio q/ω and is not always zero. However, the system remains in the Fermi liquid state in the sense that the renormalization factor Z is finite at the Fermi surface.