Ground-State Spaces of Frustration-Free Hamiltonians

We study the ground-state space properties for frustration-free Hamiltonians. We introduce a concept of `reduced spaces' to characterize local structures of ground-state spaces. For a many-body system, we characterize mathematical structures for the set Θ_k of all the k-particle reduced spaces, which with a binary operation called join forms a semilattice that can be interpreted as an abstract convex structure. The smallest nonzero elements in Θ_k, called atoms, are analogs of extreme points. We study the properties of atoms in Θ_k and discuss its relationship with ground states of k-local frustration-free Hamiltonians. For spin-1/2 systems, we show that all the atoms in Θ₂ are unique ground states of some 2-local frustration-free Hamiltonians. Moreover, we show that the elements in Θ_k may not be the join of atoms, indicating a richer structure for Θ_k beyond the convex structure. Our study of Θ_k deepens the understanding of ground-state space properties for frustration-free Hamiltonians, from a new angle of reduced spaces.