  The quantum Hall effect (QHE) originates from discrete Landau levels forming
in a two-dimensional (2D) electron system in a magnetic field. 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. 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. 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. High-quality graphite
films present a novel QHE system with a parity-controlled 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.
