Symmetric estimator for discrete self-energy of discrete many-body systems

We derive a discrete spectral representation of the single-particle self-energy using a discrete evaluation of Kugler's symmetric improved estimator. Our construction can be used on both the real and the complex (Matsubara) frequency axis. It is guaranteed to remain causal at the numerical level, in contrast to standard approaches that may generate unphysical negative spectral weight or require additional broadening. Our representation can be used for any Hamiltonian; here we apply it to quantum impurity models and in dynamical mean-field theory. The latter is formulated with a discrete hybridization function throughout its self-consistency loop. In both cases and across various numerical methods, we obtain significantly improved accuracy for a range of impurity properties.