  We introduce the concept of self-energy dispersion as an error bound on local
theories and apply it to the two-dimensional Hubbard model on the square
lattice at half-filling. Since the self-energy has no single-particle analog
and is not directly measurable in experiments, its general behavior as a
function of momentum is an open question. In this article we benchmark the
momentum dependence with the Two-Particle Self-Consistent approach together
with analytical and numerical considerations and we show that through the
addition of a local single-particle potential to the Hubbard model the
self-energy can be flattened, such that it is essentially described by only a
frequency-dependent term. We use this observation to motivate that local
theories, such as the dynamical mean-field theory, should be expected to give
very accurate results in the presence of a potential of this kind. Finally, we
propose a simple energy argument as an estimator for the crossover from
non-local to local self-energies, which can be computed even by local theories
such as dynamical mean-field theory.
