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 Theta(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 Theta(k), called atoms, are analogs of extreme points. We study the properties of atoms in Theta(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 Theta(2) are unique ground states of some 2-local frustration-free Hamiltonians. Moreover, we show that the elements in Theta(k) may not be the join of atoms, indicating a richer structure for Theta(k) beyond the convex structure. Our study of Theta(k) deepens the understanding of ground-state space properties for frustration-free Hamiltonians, from the new perspective of reduced spaces. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4748527]