  Quantum systems are often described by parameter-dependent Hamiltonians.
Points in parameter space where two levels are degenerate can carry a
topological charge. Here we theoretically study an interacting two-spin system
where the degeneracy points form a nodal loop or a nodal surface in the
magnetic parameter space, similarly to such structures discovered in the band
structure of topological semimetals. We determine the topological charge
distribution along these degeneracy geometries. We show that these
non-point-like degeneracy patterns can be obtained not only by fine-tuning, but
they can be stabilized by spatial symmetries. Since simple spin systems such as
the one studied here are ubiquitous in condensed-matter setups, we expect that
our findings, and the physical consequences of these nontrivial degeneracy
geometries, are testable in experiments with quantum dots, molecular magnets,
and adatoms on metallic surfaces.
