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The optical conductivity contains information about energy absorption and theunderlying physical processes. In finite-dimensional systems, vertexcorrections to the bare bubble need to be considered, which is acomputationally challenging task. Recent numerical studies showed that in theweak coupling limit, near an ordering instability with wave vector $\pi$,$\pi$-tons (or Maki-Thompson diagrams) yield the most relevant vertexcorrections. This provides a route for including vertex corrections into, forexample, dynamical mean field theory estimates of the optical conductivity. Byimplementing calculations on the Kadanoff-Baym contour, we reveal thecharacteristic spectral signatures of the $\pi$-tons and their evolution undernon-equilibrium conditions. We consider interaction quenches of theweakly-correlated Hubbard model near the antiferromagnetic phase boundary, andanalyze the evolution of the Drude and $\pi$-ton features. While the bubblecontribution to the optical conductivity is found to thermalize rapidly, aftersome oscillations with frequencies related to the local spectral function, the$\pi$-ton contribution exhibits a slower evolution. We link this observation tothe prethermalization phenomenon which has been previously studied in weaklyinteracting, quenched Hubbard models.

Non-equilibrium evolution of the optical conductivity of the weakly interacting Hubbard model: Drude response and $\pi$-ton type vertex corrections