Universal transport at Lifshitz metal-insulator transitions in two dimensions

We study the charge transport across a band-tuned metal-insulator transition in two dimensions. For high temperatures $T$ and chemical potentials $\mu$ far from the transition point, conduction is ballistic and the resistance $R(T)$ verifies a simple one-parameter scaling relation. Here, we explore the limits of this semi-classical behaviour and study the quantum regime beyond, where scaling breaks down. We derive an analytical formula for the simplest Feynman diagram of the linear-response conductivity $\sigma=1/R$ of a parabolic band endowed with a finite lifetime. Our formula shows excellent agreement for experiments for a field-tuned MoTe$_2$/WSe$_2$ moir\'e bilayer, and can capture the quantum effects responsible for breaking the one-parameter scaling. We go on to discuss a fascinating prediction of our model: The resistance at the quantum-critical band-tuned Lifshitz point ($\mu=T=0$) has the universal value, $R_L=(2 \pi h)/e^2$, per degree of freedom and this value is found to be compatible with experiment. Furthermore, we investigate whether two dimensional metal-insulator transitions driven by strong electron correlations or disorder can also be classified by their quantum-critical resistance and find that $R_L$ may be useful in identifying predominantly interaction driven transitions.