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Magnetic freeze-out and anomalous Hall effect in ZrTe$_5$

The ultra-quantum limit is achieved when a magnetic field confines an electron gas in its lowest spin-polarised Landau level. Here we show that in this limit, electron doped ZrTe$_5$ shows a metal-insulator transition followed by a sign change of the Hall and Seebeck effects at low temperature. We attribute this transition to a magnetic freeze-out of charge carriers on the ionised impurities. The reduction of the charge carrier density gives way to an anomalous Hall response of the spin-polarised electrons. This behaviour, at odds with the usual magnetic freeze-out scenario, occurs in this Dirac metal because of its tiny Fermi energy, extremely narrow band gap and a large $g$-factor. We discuss the different possible sources (intrinsic or extrinsic) for this anomalous Hall contribution.