  Two hundred years ago, Andr\'e-Marie Amp\`ere discovered that electric loops
in which currents of electrons are generated by a penetrating magnetic field
can interact with each other. Here we show that Amp\`ere's observation can be
transferred to the quantum realm of interactions between triangular plaquettes
of spins on a lattice, where the electrical currents at the atomic scale are
associated with a peculiar type of the orbital motion of electrons in response
to the non-coplanarity of neighbouring spins playing the role of a magnetic
field. The resulting topological orbital moment underlies the relation of the
orbital dynamics with the topology of the spin structure. We demonstrate that
the interactions of the topological orbital moments with each other and with
the spins of the underlying lattice give rise to a new class of magnetic
interactions $-$ topological chiral interactions $-$ which can dominate over
the celebrated Dzyaloshinskii-Moriya interaction, thus opening a path for the
realization of new classes of chiral magnetic materials with three-dimensional
magnetization textures such as magnetic hopfions.
