Proton transfer through hydrogen bonds plays a fundamental role in many physical, chemical and biological processes. Proton dynamics is susceptible to quantum tunnelling, which typically involves many hydrogen bonds simultaneously, leading to correlated many-body tunnelling. In contrast to the well-studied incoherent single-particle tunnelling, our understanding of many-body tunnelling is still in its infancy. Here we report the real-space observation of concerted proton tunnelling in a cyclic water tetramer using a cryogenic scanning tunnelling microscope. This is achieved by monitoring the reversible interconversion of the hydrogen-bonding chirality of the water tetramer with a chlorine-terminated scanning tunnelling microscope tip. We found that the presence of the Cl anion at the tip apex may either enhance or suppress the concerted tunnelling process, depending on the details of the coupling symmetry between the Cl ion and the protons. Our work opens up the possibility of controlling the quantum states of protons with atomic-scale precision.

We construct a density functional theory for two-dimensional electron (hole) gases subjected to both strong magnetic fields and external potentials. In particular, we are focused on regimes near even-denominator filling factors, in which the systems form composite fermion liquids. Our theory provides a systematic and rigorous approach to determine the properties of ground states in a fractional quantum Hall regime that is modified by artificial structures. We also propose a practical way to construct an approximated functional.