  The ground spaces of a vector space of hermitian matrices, partially ordered
by inclusion, form a lattice constructible from top to bottom in terms of
intersections of maximal ground spaces. In this paper we characterize the
lattice elements and the maximal lattice elements within the set of all
subspaces using constraints on operator cones. Our results contribute to the
geometry of quantum marginals, as their lattices of exposed faces are
isomorphic to the lattices of ground spaces of local Hamiltonians.
