Momentum structure of the self-energy and its parametrization for the two-dimensional Hubbard model

Abstract

We compute the self-energy for the half-filled Hubbard model on a square lattice using lattice quantum Monte Carlo simulations and the dynamical vertex approximation. The self-energy is strongly momentum-dependent, but it can be parametrized via the noninteracting energy-momentum dispersion ${\ensuremath{\varepsilon}}_{\mathbf{k}}$, except for pseudogap features right at the Fermi edge. That is, it can be written as $\mathrm{\ensuremath{\Sigma}}({\ensuremath{\varepsilon}}_{\mathbf{k}},\ensuremath{\omega})$, with two energylike parameters $(\ensuremath{\varepsilon}, \ensuremath{\omega})$ instead of three (${k}_{x}, {k}_{y}$, and $\ensuremath{\omega}$). The self-energy has two rather broad and weakly dispersing high-energy features and a sharp $\ensuremath{\omega}={\ensuremath{\varepsilon}}_{\mathbf{k}}$ feature at high temperatures, which turns to $\ensuremath{\omega}=\ensuremath{-}{\ensuremath{\varepsilon}}_{\mathbf{k}}$ at low temperatures. Altogether this yields a $\mathcal{Z}$- and reversed-$\mathcal{Z}$-like structure, respectively, for the imaginary part of $\mathrm{\ensuremath{\Sigma}}({\ensuremath{\varepsilon}}_{\mathbf{k}},\ensuremath{\omega})$. We attribute the change of the low-energy structure to antiferromagnetic spin fluctuations.