Reduced Density Matrices and Moduli of Many-Body Eigenstates

Many-body wavefunctions usually lie in high-dimensional Hilbert spaces. However, physically relevant states, i.e, the eigenstates of the Schr\"odinger equation are rare. For many-body systems involving only pairwise interactions, these eigenstates form a low-dimensional subspace of the entire Hilbert space. The geometry of this subspace, which we call the eigenstate moduli problem is parameterized by a set of 2-particle Hamiltonian. This problem is closely related to the $N$-representability conditions for 2-reduced density matrices, a long-standing challenge for quantum many-body systems. Despite its importance, the eigenstate moduli problem remains largely unexplored in the literature. In this Letter, we propose a comprehensive approach to this problem. We discover an explicit set of algebraic equations that fully determine the eigenstate spaces of $m$-interaction systems as projective varieties, which in turn determine the geometry of the spaces for representable reduced density matrices. We investigate the geometrical structure of these spaces, and validate our results numerically using the exact diagonalization method. Finally, we generalize our approach to the moduli problem of the arbitrary family of Hamiltonians parameterized by a set of real variables.