  We present momentum-resolved spectral functions and band gaps from bare and
self-consistent second-order perturbation theory for insulating periodic
solids. We establish that, for systems with large gap sizes, both bare and
self-consistent perturbation theory yield reasonable gaps. However, smaller gap
sizes require a self-consistent adjustment of the propagator. In contrast to
results obtained within a quasiparticle formalism used on top of bare
second-order perturbation theory, no unphysical behavior of the band gap is
observed. Our implementation of a fully self-consistent, $Phi$-derivable, and
thermodynamically consistent finite-temperature diagrammatic perturbation
theory forms a framework on which embedding theories such as the dynamical
mean-field theory and self-energy embedding theories can be implemented.
