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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 $Σ(\varepsilon_{\mathbf{k}},ω)$, with two energy-like parameters ($\varepsilon$, $ω$) instead of three ($k_x$, $k_y$ and $ω$). The self-energy has two rather broad and weakly dispersing high energy features and a sharp $ω= \varepsilon_{\mathbf{k}}$ feature at high temperatures, which turns to $ω= -\varepsilon_{\mathbf{k}}$ at low temperatures. Altogether this yields a Z- and reversed-Z-like structure, respectively, for the imaginary part of $Σ(\varepsilon_{\mathbf{k}},ω)$. We attribute the change of the low energy structure to antiferromagnetic spin fluctuations.