Weak localization and crossover from Lifshitz transition in two dimensions

Dirac point plays a crucial role in regulating electronic properties of topological semimetals. In two dimensions, the manipulation of Dirac points can spur a transition from Dirac semimetal through semi-Dirac phase to a gapped phase. Across such a so-called Lifshitz transition, we find that the quantum interference corrections to the conductivity $\ensuremath{\delta}{\ensuremath{\sigma}}_xx$ and $\ensuremath{\delta}{\ensuremath{\sigma}}_yy$ are always negative, giving rise to a weak localization behavior. The ratio $\ensuremath{\delta}{\ensuremath{\sigma}}_xx/\ensuremath{\delta}{\ensuremath{\sigma}}_yy$ undergoes a transition from linear to parabolic dependence on the merging parameter across the Lifshitz transition, which leads to a crossover of the temperature dependence of the inverse inelastic scattering time $1/{\ensuremath{\tau}}_{\ensuremath{\varepsilon}}$ from $\ensuremath{\sim}T$ to $\ensuremath{\sim}Tln({T}₀/T)$. This fingerprint behavior can be readily tested experimentally through merging Dirac points in two-dimensional lattices. This work presents an alternative perspective to understand weak localization through Lifshitz transition.