Invariants of the single impurity Anderson model and implications for conductance functionals

Abstract: An exact relation between the conductance maximum $G_0$ at zero temperature and a ratio of lead densities is derived within the framework of the single impurity Anderson model: $G_0={\mathfrak R}[n] \frac{2e^2}{h}$, where ${\mathfrak R}[n]=4\Delta N_{{\cal L},x} \Delta N_{{\cal R},x}/(\Delta N_{{\cal L},x}+\Delta N_{{\cal R},x})^2$ and $\Delta N_{{\cal L},x}$, $\Delta N_{{\cal R},x}$ denote the excess density in the left/right lead at distance $x$ due to the presence of the impurity at the origin, $x=0$. The relation constitutes a parameter-free expression of the conductance of the model in terms of the ground state density that generalizes an earlier result to the generic case of asymmetric lead couplings. It turns out that the specific density ratio, ${\mathfrak R}[n]$, is independent of the distance to the impurity $x$, the (magnetic) band-structure and filling fraction of the contacting wires, the strength of the onsite interaction, the gate voltage and the temperature. Disorder induced backscattering in the contacting wires has an impact on ${\mathfrak R}$ that we discuss. Our result suggests that it should be possible, in principle, to determine experimentally the peak conductance of the Anderson impurity by performing a combination of measurements of ground-state densities.