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 $T\rightarrow T_c$, the critical temperature for antiferromagnetic or charge ordering, the dc vertex correction is algebraic $σ_{VERT}^{dc}\propto ξ\sim (T-T_c)^{-ν}$ in one dimension and logarithmic $σ_{VERT}^{dc}\propto \lnξ\sim ν\ln (T-T_c)$ in two dimensions. Here, $ν$ is the critical exponent for the correlation length. If we have the exponential scaling $ξ\sim e^{1/T}$ 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: $ω_{DDP} \sim 1/τ$. These characteristic dependencies can guide experiments to evidence $π$-tons in actual materials.