Analytical expression for π-ton vertex contributions to the optical conductivity

Abstract

Vertex corrections from the transversal particle-hole channel, so-called π-tons, are generic in models for strongly correlated electron systems and can lead to a displaced Drude peak (DDP). Here, we derive the analytical expression for these π-tons, and how they affect the optical conductivity as a function of correlation length ξ, fermion lifetime τ, temperature T, and coupling strength to spin or charge fluctuations g. In particular, for T → T<inf>c</inf>, the critical temperature for antiferromagnetic or charge ordering, the dc vertex correction is algebraic σ<inf>VERT</inf>^dc ∝ ξ ∼ (T − T<inf>c</inf>)^−ν in one dimension and logarithmic σ<inf>VERT</inf>^dc ∝ ln ξ ∼ νln(T − T<inf>c</inf>) in two dimensions. Here, ν is the critical exponent for the correlation length. If we have the exponential scaling ξ ∼ e^1/T of an ideal two-dimensional system, the DDP becomes more pronounced with increasing T but fades away at low temperatures where only a broadening of the Drude peak remains, as it is observed experimentally, with the dc resistivity exhibiting a linear T dependence at low temperatures. Further, we find the maximum of the DPP to be given by the inverse lifetime: ω<inf>DDP</inf> ∼ 1/τ. These characteristic dependencies can guide experiments to evidence π-tons in actual materials.