  The Haldane model on the honeycomb lattice is a paradigmatic example of a
Hamiltonian featuring topologically distinct phases of matter. It describes a
mechanism through which a quantum Hall effect can appear as an intrinsic
property of a band-structure, rather than being caused by an external magnetic
field. Although an implementation in a material was considered unlikely, it has
provided the conceptual basis for theoretical and experimental research
exploring topological insulators and superconductors. Here we report on the
experimental realisation of the Haldane model and the characterisation of its
topological band-structure, using ultracold fermionic atoms in a periodically
modulated optical honeycomb lattice. The model is based on breaking
time-reversal symmetry as well as inversion symmetry. The former is achieved
through the introduction of complex next-nearest-neighbour tunnelling terms,
which we induce through circular modulation of the lattice position. For the
latter, we create an energy offset between neighbouring sites. Breaking either
of these symmetries opens a gap in the band-structure, which is probed using
momentum-resolved interband transitions. We explore the resulting
Berry-curvatures of the lowest band by applying a constant force to the atoms
and find orthogonal drifts analogous to a Hall current. The competition between
both broken symmetries gives rise to a transition between topologically
distinct regimes. By identifying the vanishing gap at a single Dirac point, we
map out this transition line experimentally and quantitatively compare it to
calculations using Floquet theory without free parameters. We verify that our
approach, which allows for dynamically tuning topological properties, is
suitable even for interacting fermionic systems. Furthermore, we propose a
direct extension to realise spin-dependent topological Hamiltonians.
