The Hall number across a van Hove singularity

In the context of the relaxation time approximation to Boltzmann transport theory, we examine the behavior of the Hall number, $n_H$, of a metal in the neighborhood of a Lifshitz transition from a closed Fermi surface to open sheets. A non-analytic dependence of $n_H$ on the electron density is universal in the high field limit, but at low fields the behavior is non-singular and non-universal. We find, however, that for suitable choice of band-parameters a singular change in the low-field $n_H$ occurs near a continuous nematic-order-driven Lifshitz transition. This behavior of $n_H$ is similar to that seen in recent experiments in the high temperature superconductor YBa$_2$Cu$_3$O$_{7-x}$, where a sharp drop in $n_H$ occurs below optimal doping.