  Atomically thin two-dimensional materials such as graphene and hexagonal
boron nitride have recently been found to exhibit appreciable permeability to
thermal protons, making these materials emerging candidates for separation
technologies [S. Hu et al., Nature 516, 227 (2014); M. Lozada-Hidalgo et al.,
Science 351, 68 (2016).]. These remarkable findings remain unexplained by
density-functional electronic structure calculations, which instead yield
barriers that exceed by 1.0 eV those found in experiments. Here we resolve this
puzzle by demonstrating that the proton transfer through pristine graphene is
driven by quantum nuclear effects, which substantially reduce the transport
barrier by up to 1.4 eV compared to the results of classical molecular dynamics
(MD). Our Feynman-Kac path-integral MD simulations unambiguously reveal the
quantum tunneling mechanism of strongly interacting hydrogen ions through
two-dimensional materials. In addition, we predict a strong isotope effect of 1
eV on the transport barrier for graphene in vacuum and at room temperature.
These findings not only shed light on the graphene permeability to protons and
deuterons, but also offer new insights for controlling the underlying quantum
ion transport mechanisms in nanostructured separation membranes.
