Displaced Drude peak from <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>π</mml:mi></mml:math>-ton vertex corrections

Correlated electron systems often show strong bosonic fluctuations, e.g., of antiferromagnetic nature, around a large wave vector such as $\mathbf{q}=(\ensuremath{\pi},\ensuremath{\pi}...)$. These fluctuations can give rise to vertex corrections to the optical conductivity through the (transversal) particle-hole channel, coined $\ensuremath{\pi}$-ton contributions. Previous numerical results differed qualitatively on how such vertex corrections alter the optical conductivity. Here we clarify that $\ensuremath{\pi}$-ton vertex corrections lead to a displaced Drude peak, by studying the weakly correlated regime of a one-dimensional single-orbital Hubbard model. The proximity and enhancement of the effect when approaching a phase transition of, e.g., antiferromagnetic nature can be utilized for discriminating $\ensuremath{\pi}$-tons in experiments from other physics leading to a displaced Drude peak.