Displaced Drude peak from $\pi$-ton vertex corrections

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