The quantum Hall effect (QHE) originates from discrete Landau levels forming in a two-dimensional (2D) electron system in a magnetic field 1 . In three dimensions (3D), the QHE is forbidden because the third dimension spreads Landau levels into multiple overlapping bands, destroying the quantisation. Here we report the QHE in graphite crystals that are up to hundreds of atomic layers thick -thickness at which graphite was believed to behave as a 3D bulk semimetal 2 . We attribute the observation to a dimensional reduction of electron dynamics in high magnetic fields, such that the electron spectrum remains continuous only in the direction of the magnetic field, and only the last two quasi-one-dimensional (1D) Landau bands cross the Fermi level 3,4 . In sufficiently thin graphite films, the formation of standing waves breaks these 1D bands into a discrete spectrum, giving rise to a multitude of quantum Hall plateaux. Despite a large number of layers, we observe a profound difference between films with even and odd numbers of graphene layers. For odd numbers, the absence of inversion symmetry causes valley polarisation of the standing-wave states within 1D Landau bands. This reduces QHE gaps, as compared to films of similar thicknesses but with even layer numbers because the latter retain the inversion symmetry characteristic of bilayer graphene 5,6 . High-quality graphite films present a
novel QHE system with a paritycontrolled valley polarisation and intricate interplay between orbital, spin and valley states, and clear signatures of electron-electron interactions including the fractional QHE below 0.5 K.The Lorentz force imposed by a magnetic field, B, changes the straight ballistic motion of electrons into spiral trajectories aligned along B (Fig. 1a). Such spiral motion gives rise to Landau bands that are characterised by plane waves propagating along the B direction but quantised in the directions perpendicular to the magnetic field 7 . The increase in B changes the distance between Landau bands, and the resulting crossings of the Fermi level, EF, with the Landau band edges lead to quantum oscillations. When only the lowest few Landau bands cross EF (ultra-quantum regime), the magnetic field effectively makes the electron motion one-dimensional (with conductivity allowed only in the direction parallel to B). In normal metals such as, e.g., copper, this dimensional reduction would require B > 10,000 T. In semimetals with their small Fermi surfaces (e.g., graphite 2,4,8 ) the dimensional reduction and the ultra-quantum regime (UQR) can be reached in moderate B. Indeed, magnetotransport measurements in bulk crystals 4,9-11 and films of graphite 12,13 revealed quantum oscillations reaching the lowest Landau bands. However, no dissipationless QHE transport could be observed as expected for the 3D systems.
