Ground-state spaces of frustration-free Hamiltonians

Abstract

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 Θk 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 Θk, called atoms, are analogs of extreme points. We study the properties of atoms in Θk 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 Θk may not be the join of atoms, indicating a richer structure for Θk beyond the convex structure. Our study of Θk deepens the understanding of ground-state space properties for frustration-free Hamiltonians, from the new perspective of reduced spaces. © 2012 American Institute of Physics.