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

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 non-interacting energy-momentum dispersion $\varepsilon_{\mathbf{k}}$, except for pseudogap features right at the Fermi edge. That is, it can be written as $\Sigma(\varepsilon_{\mathbf{k}},\omega)$, with two energy-like parameters ($\varepsilon$, $\omega$) instead of three ($k_x$, $k_y$ and $\omega$). The self-energy has two rather broad and weakly dispersing high energy features and a sharp $\omega= \varepsilon_{\mathbf{k}}$ feature at high temperatures, which turns to $\omega= -\varepsilon_{\mathbf{k}}$ at low temperatures. Altogether this yields a Z- and reversed-Z-like structure, respectively, for the imaginary part of $\Sigma(\varepsilon_{\mathbf{k}},\omega)$. We attribute the change of the low energy structure to antiferromagnetic spin fluctuations.